Functions and Function Notation — ACT Math Guide

Q: What You Need to Know

• A function assigns exactly one output (y-value) to each input (x-value) • Function notation f(x) means "f of x" — the output when x is the input • To evaluate f(3), substitute 3 for every x in the function rule • The domain is all possible input values; the range is all possible output values • Co

Functions and function notation ACT questions appear regularly throughout the math section, testing your ability to work with mathematical relationships. Functions are simply rules that assign exactly one output to each input, like a machine that transforms numbers. The ACT math section includes 8-12 function questions among its 60 questions in 60 minutes, making this topic crucial for your score. With the right approach, function problems become straightforward point-grabbers that boost your confidence.

What You Need to Know

A function assigns exactly one output (y-value) to each input (x-value)

Function notation f(x) means "f of x" — the output when x is the input

To evaluate f(3), substitute 3 for every x in the function rule

The domain is all possible input values; the range is all possible output values

Composite functions like f(g(x)) work from the inside out

Inverse functions undo what the original function does

📐 KEY FORMULA: f(x) = [rule] means substitute the input for x

⏱️ ACT TIME TIP: When you see f(2), immediately substitute 2 for every x — don't overthink it

How to Solve Functions and Function Notation on the ACT

Example Question 1 — Easy/Medium Difficulty

If f(x) = 2x² - 3x + 1, what is f(4)?

A) 17

B) 19

C) 21

D) 23

E) 25

Solution:

Step 1: Substitute 4 for every x in the function

Step 2: f(4) = 2(4)² - 3(4) + 1

Step 3: f(4) = 2(16) - 12 + 1 = 32 - 12 + 1 = 21

✅Answer: C — Direct substitution gives us 21 when we replace x with 4.

Example Question 2 — Hard Difficulty

If g(x) = x² + 2x and h(x) = 3x - 1, what is g(h(2))?

A) 15

B) 21

C) 35

D) 45

E) 55

Solution:

Step 1: Work from the inside out — find h(2) first

Step 2: h(2) = 3(2) - 1 = 6 - 1 = 5

Step 3: Now find g(5) = (5)² + 2(5) = 25 + 10 = 35

✅Answer: C — Composite functions require working from the innermost function outward.

Common ACT Math Mistakes to Avoid

❌Mistake: Confusing f(x + 2) with f(x) + 2

✅Fix: f(x + 2) means substitute (x + 2) for every x in the original function

❌Mistake: Forgetting order of operations when evaluating functions

✅Fix: Follow PEMDAS carefully — exponents before multiplication, multiplication before addition

❌Mistake: Working composite functions from left to right instead of inside out

✅Fix: For f(g(x)), always evaluate g(x) first, then substitute that result into f

❌Mistake: Mixing up domain and range definitions

✅Fix: Domain = inputs (x-values), Range = outputs (y-values)

Practice Question — Try It Yourself

If f(x) = x³ - 2x + 5, what is f(-2)?

A) -15

B) -7

C) 1

D) 9

E) 17

Show Answer

Answer: C — f(-2) = (-2)³ - 2(-2) + 5 = -8 + 4 + 5 = 1

Key Takeaways for the ACT

Function notation f(x) just means "substitute x into the rule"

Always work composite functions from the inside parentheses outward

Your calculator handles the arithmetic — focus on correct substitution

ACT math functions often involve quadratics, so watch your order of operations

Practice evaluating functions quickly since you have one minute per question

The five answer choices (A through E) help you check your work