Understanding Function Notation & Evaluating Functions | ACT Math Guide for Grades 9-12
If you’ve ever wondered what $$f(x)$$ really means or why functions matter for your ACT Math score, you’re in the right place! Functions are one of the most tested topics in the ACT Math section, appearing in 12-15% of all questions. That’s roughly 7-9 questions out of 60, making this topic absolutely crucial for your composite score. [[2]](#__2)
ACT SCORE BOOSTER: Master This Topic for 3-5 Extra Points!
Functions appear in 12-15% of ACT Math questions (7-9 questions per test). Understanding function notation and evaluation thoroughly can add 3-5 points to your composite score. These are some of the most straightforward points you can earn with the right strategies! [[2]](#__2)
🚀 Jump to ACT Strategy →📚 What Are Functions and Why Do They Matter?
A function is simply a mathematical relationship where each input produces exactly one output. Think of it like a vending machine: you press a button (input), and you get a specific snack (output). Every time you press B3, you get the same chips—that’s what makes it a function! [[3]](#__3)
In the ACT Math section, functions are tested extensively because they form the foundation for advanced mathematics, including calculus and statistics that you’ll encounter in college. The ACT specifically tests your ability to understand function notation (like $$f(x)$$), evaluate functions by substituting values, and interpret what functions mean in real-world contexts. [[0]](#__0)
Why it matters for your ACT score: Function questions are often among the quickest to solve once you understand the pattern. While geometry problems might take 90 seconds, a well-prepared student can solve function evaluation problems in 30-45 seconds, giving you more time for challenging questions. [[1]](#__1)
📐 Key Concepts & Function Notation Rules
🔹 Understanding Function Notation
$$f(x)$$ is read as “f of x” and means “the function f evaluated at x”
- $$f$$ = the name of the function (could be $$g$$, $$h$$, or any letter)
- $$x$$ = the input variable (independent variable)
- $$f(x)$$ = the output value (dependent variable)
💡 Important: $$f(x)$$ does NOT mean “f times x”—it’s a notation showing the relationship between input and output!
🔹 Evaluating Functions: The Substitution Method
To evaluate $$f(a)$$, replace every $$x$$ in the function with the value $$a$$
Example Format:
If $$f(x) = 2x + 3$$, then:
$$f(5) = 2(5) + 3 = 10 + 3 = 13$$
🔹 Common Function Types on the ACT
| Function Type | General Form | Example |
|---|---|---|
| Linear | $$f(x) = mx + b$$ | $$f(x) = 3x – 2$$ |
| Quadratic | $$f(x) = ax^2 + bx + c$$ | $$f(x) = x^2 + 4x – 5$$ |
| Absolute Value | $$f(x) = |x|$$ | $$f(x) = |2x – 3|$$ |
| Piecewise | Different rules for different inputs | $$f(x) = \begin{cases} x+1 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$$ |
🌍 Real-World Applications of Functions
📱 Cell Phone Plans
If your phone plan charges $$30 + 0.10$$ per text message, the function is $$C(t) = 30 + 0.10t$$, where $$t$$ is the number of texts. To find the cost for 50 texts: $$C(50) = 30 + 0.10(50) = 30 + 5 = 35$$ dollars.
🚗 Uber/Lyft Pricing
A ride-share service charges a base fee of $$2.50$$ plus $$1.75$$ per mile. The function is $$P(m) = 2.50 + 1.75m$$. For a 12-mile trip: $$P(12) = 2.50 + 1.75(12) = 2.50 + 21 = 23.50$$ dollars.
🏃♂️ Fitness & Calorie Burning
If you burn 8 calories per minute running, the function is $$C(t) = 8t$$, where $$t$$ is time in minutes. After 45 minutes: $$C(45) = 8(45) = 360$$ calories burned.
💰 Salary & Commission
A salesperson earns $$2000$$ base salary plus $$150$$ per sale. The function is $$S(n) = 2000 + 150n$$. For 18 sales: $$S(18) = 2000 + 150(18) = 2000 + 2700 = 4700$$ dollars.
ACT Connection: The ACT frequently uses real-world scenarios like these to test your understanding of functions. Being able to translate word problems into function notation is a critical skill! [[0]](#__0)
✅ Step-by-Step Examples with Solutions
📌 Example 1: Basic Function Evaluation
If $$f(x) = 3x – 7$$, find $$f(4)$$.
Step 1: Identify the function and the input value
Function: $$f(x) = 3x – 7$$
Input: $$x = 4$$
Step 2: Replace every $$x$$ with 4
$$f(4) = 3(4) – 7$$
Step 3: Simplify using order of operations
$$f(4) = 12 – 7$$
$$f(4) = 5$$
✅ Final Answer: $$f(4) = 5$$
⏱️ ACT Time Estimate: 20-30 seconds
📌 Example 2: Quadratic Function Evaluation
If $$g(x) = x^2 – 5x + 6$$, find $$g(-3)$$.
Step 1: Write out the function with the input value
$$g(-3) = (-3)^2 – 5(-3) + 6$$
⚠️ Common Mistake Alert: When substituting negative numbers, always use parentheses! $$(-3)^2 = 9$$, not $$-9$$.
Step 2: Calculate each term separately
$$(-3)^2 = 9$$
$$-5(-3) = 15$$
Constant: $$6$$
Step 3: Combine all terms
$$g(-3) = 9 + 15 + 6 = 30$$
✅ Final Answer: $$g(-3) = 30$$
⏱️ ACT Time Estimate: 30-45 seconds
📌 Example 3: Function Composition
If $$f(x) = 2x + 1$$ and $$g(x) = x^2$$, find $$f(g(3))$$.
Step 1: Work from the inside out—evaluate $$g(3)$$ first
$$g(3) = 3^2 = 9$$
Step 2: Use that result as the input for $$f(x)$$
Now we need to find $$f(9)$$
Step 3: Evaluate $$f(9)$$
$$f(9) = 2(9) + 1 = 18 + 1 = 19$$
✅ Final Answer: $$f(g(3)) = 19$$
💡 Pro Tip: Function composition $$f(g(x))$$ means “apply $$g$$ first, then apply $$f$$ to the result.” Think of it like putting on socks ($$g$$) before shoes ($$f$$)! [[3]](#__3)
⏱️ ACT Time Estimate: 45-60 seconds
📌 Example 4: Real-World Application (ACT-Style)
A streaming service charges a monthly fee based on the function $$C(h) = 12 + 0.50h$$, where $$h$$ is the number of hours of premium content watched. How much will a customer pay if they watch 24 hours of premium content in one month?
Step 1: Identify what the question is asking
We need to find $$C(24)$$ (the cost when $$h = 24$$)
Step 2: Substitute $$h = 24$$ into the function
$$C(24) = 12 + 0.50(24)$$
Step 3: Calculate
$$C(24) = 12 + 12 = 24$$
Step 4: Interpret the answer in context
The customer will pay $24 for the month.
✅ Final Answer: $24.00
⏱️ ACT Time Estimate: 40-50 seconds
📝 ACT-Style Practice Questions
Test your understanding with these ACT-style practice problems. Try solving them on your own before checking the solutions! [[1]](#__1)
Practice Question 1 BASIC
If $$f(x) = 5x – 3$$, what is the value of $$f(6)$$?
Show Solution
Solution:
$$f(6) = 5(6) – 3 = 30 – 3 = 27$$
✅ Correct Answer: A) 27
Practice Question 2 INTERMEDIATE
For $$h(x) = 2x^2 – 3x + 1$$, what is $$h(-2)$$?
Show Solution
Solution:
$$h(-2) = 2(-2)^2 – 3(-2) + 1$$
$$h(-2) = 2(4) + 6 + 1$$
$$h(-2) = 8 + 6 + 1 = 15$$
✅ Correct Answer: D) 15
Key Point: Remember that $$(-2)^2 = 4$$, and $$-3(-2) = +6$$
Practice Question 3 INTERMEDIATE
If $$f(x) = x + 4$$ and $$g(x) = 3x$$, what is $$f(g(2))$$?
Show Solution
Solution:
Step 1: Evaluate $$g(2)$$ first
$$g(2) = 3(2) = 6$$
Step 2: Now evaluate $$f(6)$$
$$f(6) = 6 + 4 = 10$$
✅ Correct Answer: C) 10
Practice Question 4 ADVANCED
A taxi company charges according to the function $$C(m) = 3.50 + 2.25m$$, where $$m$$ is the number of miles traveled. If a customer’s fare was $25.75, how many miles did they travel?
Show Solution
Solution:
We know $$C(m) = 25.75$$, so:
$$3.50 + 2.25m = 25.75$$
$$2.25m = 25.75 – 3.50$$
$$2.25m = 22.25$$
$$m = 22.25 \div 2.25 = 9.889… \approx 10$$
✅ Correct Answer: C) 10 miles
ACT Strategy: This is a “reverse” function problem—you’re given the output and finding the input. Set up an equation and solve for the variable! [[0]](#__0)
Practice Question 5 ADVANCED
If $$f(x) = |2x – 5|$$, what is $$f(-3)$$?
Show Solution
Solution:
$$f(-3) = |2(-3) – 5|$$
$$f(-3) = |-6 – 5|$$
$$f(-3) = |-11|$$
$$f(-3) = 11$$
✅ Correct Answer: D) 11
Remember: Absolute value always gives a non-negative result. $$|-11| = 11$$
💡 ACT Pro Tips & Tricks for Functions
🎯 Tip #1: Use Parentheses for Negative Numbers
Always wrap negative numbers in parentheses when substituting: $$f(-3)$$ means replace $$x$$ with $$(-3)$$, not $$-3$$. This prevents sign errors, especially with exponents. $$(-3)^2 = 9$$, but $$-3^2 = -9$$. [[0]](#__0)
⚡ Tip #2: Work Inside-Out for Composition
For $$f(g(x))$$, always evaluate the inner function first ($$g$$), then use that result in the outer function ($$f$$). Think “PEMDAS”—work from the inside out, just like with parentheses in order of operations.
🔍 Tip #3: Check Your Answer with the Original Function
After finding $$f(a)$$, quickly verify by asking: “Does this output make sense given the input?” For linear functions, larger inputs should give proportionally larger outputs (if the slope is positive). [[1]](#__1)
📊 Tip #4: Recognize Common Function Patterns
Linear functions ($$mx + b$$) change at a constant rate. Quadratic functions ($$x^2$$) create parabolas. Absolute value functions ($$|x|$$) always produce non-negative outputs. Recognizing these patterns helps you eliminate wrong answers quickly.
⏱️ Tip #5: Calculator Strategy for Complex Functions
For complicated functions, use your calculator’s “Y=” function. Enter the function as Y1, then evaluate by typing Y1(value). This saves time and reduces arithmetic errors on test day. [[0]](#__0)
🚫 Tip #6: Avoid the “Multiplication” Trap
$$f(x)$$ does NOT mean “$$f$$ times $$x$$”! This is the #1 misconception. $$f(x)$$ is notation showing the relationship between input and output. If you see $$f(3)$$, you’re evaluating the function at $$x = 3$$, not multiplying.
🚫 Common Mistakes to Avoid
❌ Mistake #1: Sign Errors with Negative Inputs
Wrong: For $$f(x) = x^2 – 4$$, evaluating $$f(-2)$$ as $$-2^2 – 4 = -4 – 4 = -8$$
Right: $$f(-2) = (-2)^2 – 4 = 4 – 4 = 0$$
Fix: Always use parentheses around negative numbers!
❌ Mistake #2: Order Confusion in Composition
Wrong: For $$f(g(2))$$, evaluating $$f(2)$$ first
Right: Evaluate $$g(2)$$ first, then use that result in $$f$$
Fix: Remember: work from the inside out, like nested parentheses!
❌ Mistake #3: Forgetting to Distribute
Wrong: For $$f(x) = 2(x + 3)$$, evaluating $$f(4)$$ as $$2(4) + 3 = 11$$
Right: $$f(4) = 2(4 + 3) = 2(7) = 14$$
Fix: Replace ALL instances of $$x$$ with the input value before simplifying!
🎯 ACT Test-Taking Strategy for Functions
⏱️ Time Management
Allocate 30-45 seconds for basic function evaluation questions and 60-90 seconds for composition or word problems. If you’re stuck after 60 seconds, mark it and move on—you can always return. [[0]](#__0)
🎲 Strategic Guessing
If you must guess, eliminate answers that don’t make logical sense. For example, if the function is $$f(x) = x^2$$ and you’re evaluating $$f(-3)$$, eliminate any negative answer choices since squares are always non-negative.
✅ Quick Verification Method
After solving, do a quick “reasonableness check”: If $$f(x) = 2x + 5$$ and you found $$f(10) = 100$$, that should trigger alarm bells (correct answer is 25). This 3-second check can save you from careless errors. [[1]](#__1)
🔄 When to Use Your Calculator
Use your calculator for functions with decimals, large numbers, or complex arithmetic. For simple substitutions like $$f(x) = x + 3$$, mental math is faster. Store the function in your calculator’s Y= menu for repeated evaluations—this is especially useful for composition problems.
🎯 Trap Answer Recognition
ACT test writers include common mistakes as answer choices. Watch for: (1) answers that result from sign errors with negatives, (2) answers from evaluating composition in the wrong order, and (3) answers from treating $$f(x)$$ as multiplication. If your answer matches choice A or B and seems too easy, double-check your work!
📋 The 3-Step Function Checklist
- Identify: What function? What input value?
- Substitute: Replace every $$x$$ with the input (use parentheses!)
- Simplify: Follow order of operations carefully
🧠 Memory Tricks & Mnemonics
🎯 “SIPS” Method for Function Evaluation
See the function
Identify the input
Plug it in (with parentheses!)
Simplify step by step
🔄 “Inside Before Outside” for Composition
For $$f(g(x))$$, think of Russian nesting dolls: you must open the inner doll ($$g$$) before you can see the outer one ($$f$$). Always work from the inside out!
📦 “Function Machine” Visualization
Picture a function as a machine: you drop a number in the top (input), the machine processes it according to its rule, and a new number comes out the bottom (output). This helps you remember that $$f(x)$$ is NOT multiplication—it’s a transformation process.
🎨 Visual Representation: How Functions Work
FUNCTION MACHINE: f(x) = 2x + 3 ┌─────────────────────────────────┐ │ │ │ INPUT: x = 5 │ │ ↓ │ │ ┌─────────┐ │ │ │ │ │ │ x → │ f(x) = │ → f(x) │ │ │ 2x + 3 │ │ │ │ │ │ │ └─────────┘ │ │ ↓ │ │ 2(5) + 3 = 13 │ │ ↓ │ │ OUTPUT: 13 │ │ │ └─────────────────────────────────┘
COMPOSITION: f(g(x)) where f(x) = x + 4 and g(x) = 3x Finding f(g(2)):
Step 1: Inner function first
┌──────────────┐
│ g(2) = ? │
│ g(x) = 3x │
│ g(2) = 3(2)│
│ g(2) = 6 │
└──────────────┘
↓
Step 2: Use result in outer function
┌──────────────┐
│ f(6) = ? │
│ f(x) = x+4 │
│ f(6) = 6+4 │
│ f(6) = 10 │
└──────────────┘
↓
ANSWER: f(g(2)) = 10
These visual representations help you understand the flow of function evaluation. The input goes in, gets transformed by the function’s rule, and produces an output.
❓ Frequently Asked Questions (FAQs)
Q1: What’s the difference between $$f(x)$$ and $$f \cdot x$$?
A: $$f(x)$$ is function notation meaning “the function $$f$$ evaluated at $$x$$”—it shows a relationship. $$f \cdot x$$ would mean “$$f$$ multiplied by $$x$$,” which is completely different. Function notation uses parentheses to indicate evaluation, not multiplication. This is one of the most common sources of confusion for students!
Q2: How do I know if I should use my calculator for function problems?
A: Use your calculator when: (1) the function involves decimals or fractions, (2) you need to evaluate the same function multiple times, or (3) the arithmetic is complex (like $$7.5^2 – 3.2(7.5) + 1.8$$). For simple functions like $$f(x) = x + 5$$, mental math is faster. The TI-84 calculator’s “Y=” function is particularly useful—enter the function once and evaluate it multiple times quickly.
Q3: What if the function has two variables, like $$f(x, y) = 2x + 3y$$?
A: Functions with multiple variables work the same way—just substitute each value in its correct place. For $$f(4, 5)$$, replace $$x$$ with 4 and $$y$$ with 5: $$f(4, 5) = 2(4) + 3(5) = 8 + 15 = 23$$. These appear less frequently on the ACT but follow the same substitution principle.
Q4: Can a function have the same output for different inputs?
A: Yes! A function can have the same output for different inputs. For example, $$f(x) = x^2$$ gives $$f(3) = 9$$ and $$f(-3) = 9$$. The key rule is that each INPUT must produce exactly ONE output, but multiple inputs can share the same output. This is called a “many-to-one” relationship and is perfectly valid for functions.
Q5: How can I get faster at evaluating functions on the ACT?
A: Practice these three strategies: (1) Pattern recognition—learn to quickly identify function types (linear, quadratic, etc.), (2) Mental math—strengthen your ability to calculate simple operations without writing everything down, and (3) Systematic approach—use the SIPS method (See, Identify, Plug, Simplify) every time so it becomes automatic. Students who practice 10-15 function problems daily for two weeks typically cut their solving time in half.
📈 Score Improvement Tips: From Good to Great
🎯 Target Score 20-24: Master the Basics
Focus on linear function evaluation ($$f(x) = mx + b$$) and simple substitution. These appear in 60% of function questions and are the easiest points to secure. Practice 5 basic problems daily until you can solve them in under 30 seconds each.
🎯 Target Score 25-29: Add Complexity
Master quadratic functions, absolute value functions, and basic composition ($$f(g(x))$$). Learn to recognize when to use your calculator versus mental math. Practice word problems that require translating real-world scenarios into function notation. Aim for 80% accuracy on intermediate-level problems.
🎯 Target Score 30-36: Perfect Your Strategy
Focus on advanced composition, piecewise functions, and “reverse” problems (given output, find input). Learn to spot trap answers immediately. Practice under timed conditions: 30 seconds for basic, 45 seconds for intermediate, 60 seconds for advanced. At this level, it’s not about knowing more—it’s about executing faster and more accurately.
💡 Universal Tip: The students who improve most dramatically are those who review their mistakes systematically. After each practice session, spend 5 minutes analyzing WHY you got problems wrong—was it a conceptual misunderstanding, a calculation error, or a time management issue? Address the root cause, not just the symptom.
🎓 Wrapping It Up: Your Path to Function Mastery
Understanding function notation and evaluation is one of the highest-yield topics you can master for the ACT Math section. With 12-15% of questions testing this concept, you’re looking at 7-9 questions per test—that’s potentially 3-5 points on your composite score just from this one topic!
Remember the key principles: (1) $$f(x)$$ is notation, not multiplication, (2) always use parentheses when substituting negative numbers, (3) work inside-out for composition, and (4) check your answers for reasonableness. These four rules will prevent 90% of common errors.
The real-world applications we covered—from cell phone plans to ride-share pricing—aren’t just examples; they’re the exact types of scenarios the ACT uses to test your understanding. When you can translate a word problem into function notation and evaluate it correctly, you’re demonstrating the mathematical reasoning skills that colleges value.
🚀 Practice consistently, review your mistakes, and watch your confidence—and your score—soar. You’ve got this!
✍️ Written by Irfan Mansuri
ACT Test Prep Specialist & Educator
IrfanEdu.com • United States
Irfan Mansuri is a dedicated ACT test preparation specialist with over 15 years of experience helping high school students achieve their target scores. As the founder of IrfanEdu.com, he has guided thousands of students through the ACT journey, with many achieving scores of 30+ and gaining admission to their dream colleges. His teaching methodology combines deep content knowledge with proven test-taking strategies, making complex concepts accessible and helping students build confidence. Irfan’s approach focuses not just on memorization, but on true understanding and strategic thinking that translates to higher scores.
📚 Related ACT Math Resources
- Linear Equations & Inequalities: Build your algebra foundation
- Quadratic Functions & Parabolas: Advanced function concepts
- Systems of Equations: Working with multiple functions
- Coordinate Geometry: Graphing functions on the coordinate plane
- ACT Math Time Management: Strategies for the full 60-question section
Continue building your ACT Math skills by exploring these related topics on IrfanEdu.com!
📖 Sources & References
- Piqosity. (2024). “ACT Math Strategies | Math Tips for the 2025 ACT.” Retrieved from https://www.piqosity.com/act-math-tips-strategies/
- Time Flies Education. (2024). “10 Practice Questions for the Math Portion of the ACT.” Retrieved from https://timefliesedu.com/2024/06/29/10-practice-questions-for-the-math-portion-of-the-act/
- ACT. (2024). “Preparing for the ACT® Test 2024-2025.” Retrieved from ACT Official Guide
- Fiveable. (2024). “Preparing for Higher Math: Functions – ACT Study Guide.” Retrieved from https://fiveable.me/act/math/functions/study-guide/
Understanding Functions and Function Notation
Welcome, future mathematician! Have you ever wondered how your smartphone knows exactly how much battery life you have left? Or how weather apps predict tomorrow’s temperature? The answer lies in something called functions—one of the most powerful concepts in mathematics!
Think about it: when you drive faster, you cover more distance. When you study more hours, your grades improve. When you add more ingredients, you make more cookies. In each case, one thing depends on another. That’s exactly what functions help us understand and predict!
🎯 What Exactly is a Function?
Real-World Connection
Imagine you’re at a vending machine. You press button A3, and you get a specific snack—let’s say, chips. Every time you press A3, you get chips. You never press A3 and get both chips AND a soda. That’s how functions work! One input (button press) gives you exactly one output (snack).
Before we define functions, let’s start with something simpler called a relation.
Understanding Relations First
A relation is simply a way of pairing things from one group with things from another group. Think of it like matching students with their test scores, or cities with their temperatures.
Students (Input)
Ages (Output)
✓ This IS a function! Each student has exactly ONE age.
In this relation:
- The domain (all inputs) = {Sarah, Mike, Emma, James}
- The range (all outputs) = {16, 17, 18}
- Notice that Sarah and Emma are both 16—that’s okay! Different inputs can have the same output.
Official Definition: Function
A function is a special type of relation where each input is paired with exactly one output. No input can have multiple outputs!
Think of it this way: If you know the input, you can predict exactly what the output will be—no surprises, no multiple possibilities!
- Domain: All possible input values (the $$x$$ values)
- Range: All possible output values (the $$y$$ values)
When is Something NOT a Function?
Students (Input)
Favorite Colors (Output)
✗ This is NOT a function if Alex likes both Blue AND Red!
✓ This IS a Function
Phone Number → Owner
Each phone number belongs to exactly one person. When you call a number, you reach one specific person.
✗ This is NOT a Function
Person → Phone Number
One person might have multiple phone numbers (home, mobile, work). One input, multiple outputs!
Quick Check Method
Ask yourself: “If I give you an input, can you tell me exactly one output without any doubt?”
- If YES → It’s a function! ✓
- If NO (because there could be multiple outputs) → Not a function! ✗
Example 1: Streaming Service Subscriptions
A streaming service has these subscription plans:
| Plan Name | Monthly Price |
|---|---|
| Basic | $8.99 |
| Standard | $13.99 |
| Premium | $17.99 |
Question 1: Is price a function of plan name?
Answer: YES! ✓
Why? Each plan name (Basic, Standard, Premium) has exactly ONE price. If you choose “Standard,” you pay $13.99—not multiple prices.
Question 2: Is plan name a function of price?
Answer: YES! ✓
Why? Each price corresponds to exactly ONE plan. If you’re paying $13.99, you have the Standard plan—no confusion!
Example 2: Online Shopping Delivery Times
An online store offers these delivery options:
| Delivery Speed | Cost |
|---|---|
| Standard (5-7 days) | Free |
| Express (2-3 days) | $5.99 |
| Next Day | $12.99 |
| Same Day | $12.99 |
Question: Is delivery speed a function of cost?
Answer: NO! ✗
Why? The cost $12.99 corresponds to TWO different delivery speeds (Next Day AND Same Day). One input (price) produces multiple outputs (delivery options), so it’s not a function!
Practice Problem 1
A school cafeteria has this menu:
| Food Item | Calories |
|---|---|
| Burger | 550 |
| Salad | 250 |
| Pizza Slice | 300 |
| Sandwich | 400 |
Question: Is calories a function of food item?
Answer: YES! ✓
Explanation: Each food item has exactly one calorie count. If you choose “Burger,” you know it has 550 calories—not 550 or 600 or any other number. One input → One output!
📝 Function Notation: The Mathematical Language
Now that you understand what functions are, let’s learn how to write them mathematically. Function notation is like a shorthand that mathematicians use worldwide—once you learn it, you can communicate complex ideas simply!
Why Do We Need Special Notation?
Imagine texting your friend: “The temperature in degrees Fahrenheit depends on the temperature in degrees Celsius.” That’s long! Instead, we write: $$F = f(C)$$. Much cleaner, right?
The Anatomy of Function Notation
The Function Name
Like naming a recipe
The Input
What you put in
The Output
What you get out
Important: Parentheses Don’t Mean Multiplication!
In function notation, $$f(x)$$ does NOT mean “$$f$$ times $$x$$”!
Instead, it means: “the function $$f$$ evaluated at input $$x$$” or simply “$$f$$ of $$x$$”
Think of it like: $$f(x)$$ = “What does function $$f$$ give me when I input $$x$$?”
Example 3: Temperature Conversion
Let’s say we have a function that converts Celsius to Fahrenheit. We can write:
Reading this aloud: “F is a function of C” or “Fahrenheit depends on Celsius”
The actual formula is: $$f(C) = \frac{9}{5}C + 32$$
Let’s use it!
Find $$f(0)$$: What’s 0°C in Fahrenheit?
So 0°C = 32°F (water freezes!)
Find $$f(100)$$: What’s 100°C in Fahrenheit?
So 100°C = 212°F (water boils!)
Example 4: Uber Ride Pricing
Imagine an Uber ride costs $3 base fare plus $1.50 per mile. We can write this as a function:
Where $$C$$ is the cost and $$m$$ is miles traveled.
Calculate $$f(5)$$: How much for a 5-mile ride?
Answer: A 5-mile ride costs $10.50
Calculate $$f(10)$$: How much for a 10-mile ride?
Answer: A 10-mile ride costs $18.00
Practice Problem 2
A phone plan charges $25 per month plus $0.10 per text message. Write this as a function and calculate the cost for 100 text messages.
Function: $$C = f(t) = 25 + 0.10t$$
Where $$C$$ is cost and $$t$$ is number of texts
For 100 texts:
Answer: $35.00 for the month
📊 Representing Functions with Tables
Tables are fantastic for organizing function data, especially when you have specific values to work with. They make it super easy to see the relationship between inputs and outputs at a glance!
Example 5: Social Media Followers Growth
Let’s say you’re tracking your Instagram followers over 6 months:
| Month ($$m$$) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Followers ($$f(m)$$) | 100 | 250 | 500 | 850 | 1200 | 1600 |
Reading the table:
- $$f(1) = 100$$ means: In month 1, you had 100 followers
- $$f(3) = 500$$ means: In month 3, you had 500 followers
- $$f(6) = 1600$$ means: In month 6, you had 1,600 followers
Is this a function? YES! Each month (input) has exactly one follower count (output).
How to Check if a Table Represents a Function
Check the first row (or column) for the input values
Does any input value appear more than once?
Do the repeated inputs have the SAME output? If yes → still a function! If no → NOT a function!
If each input has only one output → It’s a function! ✓
Example 6: Which Tables Show Functions?
Table A: Video Game Scores
| Player | Score |
|---|---|
| Alex | 1500 |
| Jordan | 2200 |
| Casey | 1500 |
Is this a function? YES! ✓
Each player has exactly one score. Alex and Casey both scored 1500—that’s fine! Different inputs can have the same output.
Table B: Student Course Enrollments
| Student | Course |
|---|---|
| Maria | Math |
| David | Science |
| Maria | English |
Is this a function? NO! ✗
Maria appears twice with two different courses. One input (Maria) produces multiple outputs (Math AND English), so this is NOT a function.
Practice Problem 3
Does this table represent a function?
| Hours Studied ($$h$$) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Test Score ($$s$$) | 65 | 75 | 85 | 95 |
Answer: YES! ✓ This is a function.
Why? Each input (hours studied) has exactly one output (test score). No input value repeats with different outputs.
🧮 Evaluating Functions: Finding Outputs
When you evaluate a function, you’re answering the question: “What output do I get for this specific input?” It’s like asking, “If I put this ingredient into my recipe, what will I get?”
Example 7: Evaluating a Simple Function
Given the function $$f(x) = 3x + 5$$, let’s evaluate it at different inputs:
Find $$f(2)$$:
Answer: When $$x = 2$$, the output is 11
Find $$f(0)$$:
Answer: When $$x = 0$$, the output is 5
Find $$f(-3)$$:
Answer: When $$x = -3$$, the output is -4
Example 8: Quadratic Function
Given $$g(x) = x^2 – 4x + 7$$, evaluate $$g(3)$$:
Final Answer: $$g(3) = 4$$
Practice Problem 4
Given $$h(x) = 2x^2 + 3x – 1$$, evaluate $$h(4)$$.
Solution:
Answer: $$h(4) = 43$$
🔍 Solving Functions: Finding Inputs
Sometimes we work backwards! Instead of “What output do I get?”, we ask “What input gives me this output?” This is called solving a function.
Example 9: Solving for Input
Given $$f(x) = 2x + 6$$, solve for $$f(x) = 14$$
Question: What input $$x$$ gives us an output of 14?
Answer: When $$x = 4$$, we get $$f(4) = 14$$
Check: $$f(4) = 2(4) + 6 = 8 + 6 = 14$$ ✓
Example 10: Solving a Quadratic (Two Solutions!)
Given $$g(x) = x^2 – 5x + 6$$, solve for $$g(x) = 0$$
$$x – 2 = 0$$ or $$x – 3 = 0$$
$$x = 2$$ or $$x = 3$$
Answer: Two solutions! $$x = 2$$ and $$x = 3$$ both give us $$g(x) = 0$$
Practice Problem 5
Given $$f(x) = 4x – 7$$, solve for $$f(x) = 21$$
Solution:
Answer: $$x = 7$$
Verification: $$f(7) = 4(7) – 7 = 28 – 7 = 21$$ ✓
🎨 One-to-One Functions: The Special Ones
Some functions are extra special—they’re called one-to-one functions. Not only does each input have one output, but each output comes from only one input!
One-to-One Function
A function is one-to-one if:
- Each input produces exactly one output (that’s just being a function)
- AND each output comes from exactly one input (this is the special part!)
In other words: No two different inputs can produce the same output!
✓ One-to-One Function
Student ID → Student Name
| ID: 1001 | → | Emma |
| ID: 1002 | → | Liam |
| ID: 1003 | → | Olivia |
Each ID goes to one name, and each name has one ID!
✗ NOT One-to-One
Birth Year → Age
| 2005 | → | 19 years |
| 2006 | → | 18 years |
| 2007 | → | 17 years |
It’s a function, but multiple birth years can give the same age in different years!
Example 11: Testing for One-to-One
Function A: $$f(x) = 2x + 3$$
Is it one-to-one? YES! ✓
Why? If two different inputs gave the same output:
This means the inputs must be the same! So different inputs always give different outputs.
Function B: $$g(x) = x^2$$
Is it one-to-one? NO! ✗
Why? Look at these examples:
- $$g(3) = 9$$
- $$g(-3) = 9$$
Two different inputs (3 and -3) produce the same output (9)! Not one-to-one!
Practice Problem 6
Is the function $$h(x) = x^3$$ one-to-one?
Answer: YES! ✓ It is one-to-one.
Why? Different numbers have different cubes. For example:
- $$2^3 = 8$$
- $$3^3 = 27$$
- $$(-2)^3 = -8$$ (different from $$2^3$$!)
No two different inputs produce the same output!
📈 The Vertical Line Test
The vertical line test is a super quick visual trick to check if a graph represents a function. It’s like a magic wand for identifying functions!
The Vertical Line Test
The Rule: Imagine drawing vertical lines (up and down) anywhere on a graph.
- If ANY vertical line crosses the graph more than once → NOT a function! ✗
- If EVERY vertical line crosses the graph at most once → It’s a function! ✓
Why does this work? A vertical line represents one $$x$$-value (one input). If it hits the graph twice, that means one input produces two outputs—which breaks the function rule!
✓ This IS a Function
Any vertical line crosses only once!
✗ This is NOT a Function
Vertical line crosses twice—not a function!
Quick Memory Trick
Vertical = Function Test
Think: “V for Vertical, F for Function”
If a vertical line hits more than once, it’s not a function!
↔️ The Horizontal Line Test
The horizontal line test checks if a function is one-to-one. It’s similar to the vertical line test, but we use horizontal lines instead!
The Horizontal Line Test
The Rule: Imagine drawing horizontal lines (left to right) across a function’s graph.
- If ANY horizontal line crosses the graph more than once → NOT one-to-one! ✗
- If EVERY horizontal line crosses the graph at most once → It’s one-to-one! ✓
Why? A horizontal line represents one $$y$$-value (one output). If it hits the graph twice, two different inputs produce the same output—not one-to-one!
✓ One-to-One Function
Horizontal line crosses once—one-to-one!
✗ NOT One-to-One
Horizontal line crosses twice—not one-to-one!
Remember Both Tests!
Vertical Line Test: Checks if it’s a function
Horizontal Line Test: Checks if it’s one-to-one
A graph must pass the vertical line test to be a function. If it also passes the horizontal line test, it’s a one-to-one function!
🔧 Essential Toolkit Functions
Just like a carpenter has basic tools (hammer, saw, screwdriver), mathematicians have basic “toolkit functions” that appear everywhere! Let’s meet the most important ones:
1. The Constant Function: $$f(x) = c$$
What it does: Always gives the same output, no matter what input you use!
Example: $$f(x) = 5$$
- $$f(0) = 5$$
- $$f(100) = 5$$
- $$f(-50) = 5$$
Real-world: A flat monthly subscription fee—same price every month!
2. The Identity Function: $$f(x) = x$$
What it does: Output equals input—what you put in is what you get out!
Example: $$f(x) = x$$
- $$f(3) = 3$$
- $$f(-7) = -7$$
- $$f(0) = 0$$
Real-world: Converting dollars to dollars (no conversion needed!)
3. The Absolute Value Function: $$f(x) = |x|$$
What it does: Makes everything positive (distance from zero)!
Example: $$f(x) = |x|$$
- $$f(5) = 5$$
- $$f(-5) = 5$$
- $$f(0) = 0$$
Real-world: Distance traveled (always positive, whether you go forward or backward!)
4. The Quadratic Function: $$f(x) = x^2$$
What it does: Squares the input (multiplies it by itself)!
Example: $$f(x) = x^2$$
- $$f(3) = 9$$
- $$f(-3) = 9$$
- $$f(0) = 0$$
Real-world: Area of a square with side length $$x$$, or the path of a thrown ball!
5. The Square Root Function: $$f(x) = \sqrt{x}$$
What it does: Finds what number, when squared, gives you $$x$$!
Example: $$f(x) = \sqrt{x}$$
- $$f(9) = 3$$ (because $$3^2 = 9$$)
- $$f(16) = 4$$ (because $$4^2 = 16$$)
- $$f(0) = 0$$
Important: Only works for $$x \geq 0$$ (can’t take square root of negative numbers in basic math!)
Real-world: Finding the side length of a square when you know its area!
6. The Cubic Function: $$f(x) = x^3$$
What it does: Cubes the input (multiplies it by itself three times)!
Example: $$f(x) = x^3$$
- $$f(2) = 8$$
- $$f(-2) = -8$$
- $$f(0) = 0$$
Real-world: Volume of a cube with side length $$x$$!
| Function Name | Formula | Key Feature | One-to-One? |
|---|---|---|---|
| Constant | $$f(x) = c$$ | Flat horizontal line | No (unless domain has one point) |
| Identity | $$f(x) = x$$ | Diagonal line through origin | Yes ✓ |
| Absolute Value | $$f(x) = |x|$$ | V-shaped, always positive | No |
| Quadratic | $$f(x) = x^2$$ | U-shaped parabola | No |
| Square Root | $$f(x) = \sqrt{x}$$ | Half parabola, $$x \geq 0$$ | Yes ✓ |
| Cubic | $$f(x) = x^3$$ | S-shaped curve | Yes ✓ |
Practice Problem 7: Identify the Function
Match each description to the correct toolkit function:
- The output is always 7, no matter what input you use
- When you input 4, you get 16
- When you input -5, you get 5
- When you input 3, you get 3
Answers:
- Constant function: $$f(x) = 7$$
- Quadratic function: $$f(x) = x^2$$ (because $$4^2 = 16$$)
- Absolute value function: $$f(x) = |x|$$ (because $$|-5| = 5$$)
- Identity function: $$f(x) = x$$ (output equals input)
🎓 Putting It All Together: Real-World Applications
Application 1: Cell Phone Data Plans
Your cell phone company charges $40 per month plus $10 for each GB of data over your limit.
Function: $$C(g) = 40 + 10g$$ where $$g$$ is GB over limit
Questions:
- How much do you pay if you use 3 GB over your limit?
- If your bill is $80, how many GB over did you go?
Solution 1: Evaluate $$C(3)$$
Answer: You pay $70
Solution 2: Solve $$C(g) = 80$$
Answer: You went 4 GB over your limit
Application 2: Online Shopping with Discount
An online store offers free shipping on orders over $50. For orders under $50, shipping costs $8.
| Order Amount | Shipping Cost |
|---|---|
| $30 | $8 |
| $45 | $8 |
| $50 | $0 (Free!) |
| $75 | $0 (Free!) |
| $100 | $0 (Free!) |
Question: Is shipping cost a function of order amount?
Answer: YES! ✓ Each order amount has exactly one shipping cost.
We can write this as a piecewise function:
Application 3: Fitness Tracker Calories
Your fitness tracker shows that you burn approximately 100 calories per mile when running.
Function: $$C(m) = 100m$$ where $$m$$ is miles run
| Miles Run ($$m$$) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Calories ($$C(m)$$) | 0 | 100 | 200 | 300 | 400 | 500 |
Question: How many miles do you need to run to burn 350 calories?
Solution: Solve $$C(m) = 350$$
Answer: You need to run 3.5 miles to burn 350 calories!
Key Takeaways from This Guide
- Functions are relationships where each input has exactly one output
- Function notation $$f(x)$$ is a clean way to express “$$f$$ of $$x$$”—the output when we input $$x$$
- Evaluating a function means finding the output for a given input
- Solving a function means finding the input(s) that produce a given output
- One-to-one functions have each output corresponding to exactly one input
- Vertical line test checks if a graph is a function
- Horizontal line test checks if a function is one-to-one
- Toolkit functions are the building blocks for more complex functions
- Functions are everywhere in real life—from phone bills to fitness tracking!
Final Challenge Problem
A taxi charges $4 for pickup plus $2.50 per mile.
- Write a function $$T(m)$$ for the total cost based on miles $$m$$
- Calculate the cost for a 7-mile trip
- If a trip costs $29, how many miles was it?
- Is this function one-to-one? Why or why not?
Solution 1: $$T(m) = 4 + 2.50m$$
Solution 2: Evaluate $$T(7)$$
Answer: A 7-mile trip costs $21.50
Solution 3: Solve $$T(m) = 29$$
Answer: The trip was 10 miles
Solution 4: YES, it’s one-to-one! ✓
Why? Each distance produces a unique cost, and each cost corresponds to exactly one distance. Different distances always produce different costs because we’re adding a constant amount per mile.
Understanding function notation, evaluating functions _ ACT Elementary Algebra Math Guide
Understanding function notation, evaluating functions _ ACT Elementary Algebra Math Guide
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📺 Understanding function notation, evaluating functions | ACT Elementary Algebra Math Guide
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