Function Notation and Evaluation — SAT Math Guide

Q: What You Need to Know

• f(x) means "function f of x" — x is the input, f(x) is the output • To evaluate f(3), substitute 3 for every x in the function definition • f(g(x)) is a composite function — evaluate the inner function first • Domain restrictions tell you which input values are allowed • Function notation can use

Function notation SAT questions appear in 2-3 problems on the Digital SAT, making this topic a reliable score booster in the advanced math section. Function notation is simply a way to represent relationships between variables using symbols like f(x) or g(t). These problems test your ability to substitute values, interpret function outputs, and work with composite functions. With the right approach, you can tackle even the trickiest function notation problems with confidence.

What You Need to Know

f(x) means "function f of x" — x is the input, f(x) is the output

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

f(g(x)) is a composite function — evaluate the inner function first

Domain restrictions tell you which input values are allowed

Function notation can use any letters: g(t), h(n), P(x), etc.

Piecewise functions have different rules for different input ranges

📐 KEY FORMULA: For f(x) = expression, substitute the given value for x

💡 PRO TIP: Always work from the inside out with composite functions like f(g(2))

How to Solve Function Notation Problems on the SAT

Example Question 1 — Medium Difficulty

If f(x) = 2x² - 3x + 1, what is the value of f(4)?

A) 17

B) 21

C) 25

D) 29

Solution:

Step 1: Substitute x = 4 into the function f(x) = 2x² - 3x + 1

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

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

✅Answer: B — Direct substitution gives us f(4) = 21

Example Question 2 — Hard Difficulty

Let g(x) = x² + 2x and h(x) = 3x - 1. If g(h(2)) = k, what is the value of k?

A) 35

B) 40

C) 45

D) 55

Solution:

Step 1: Find h(2) first since it's the inner function: h(2) = 3(2) - 1 = 5

Step 2: Now evaluate g(5) using g(x) = x² + 2x

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

✅Answer: A — Working inside-out, g(h(2)) = g(5) = 35

Common SAT Math Mistakes to Avoid

❌Mistake: Forgetting to substitute into all instances of the variable

✅Fix: Replace every x (or other variable) with the given value

❌Mistake: Evaluating composite functions in the wrong order

✅Fix: Always evaluate the innermost function first, then work outward

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

✅Fix: f(x + 2) means substitute (x + 2) for x; f(x) + 2 means add 2 to the function output

❌Mistake: Ignoring domain restrictions in piecewise functions

✅Fix: Check which piece of the function applies to your input value

Practice Question — Try It Yourself

If f(x) = (x - 1)/(x + 2) and f(a) = 1/2, what is the value of a?

A) -5

B) -4

C) 4

D) 5

Show Answer

Answer: C — Set (a - 1)/(a + 2) = 1/2, cross multiply to get 2(a - 1) = a + 2, which gives 2a - 2 = a + 2, so a = 4

Key Takeaways for the SAT

Function notation is just a way to organize input-output relationships

Always substitute carefully and replace every instance of the variable

For composite functions like f(g(x)), evaluate the inner function first

SAT math function notation problems often involve algebraic manipulation

Practice with different function types: linear, quadratic, rational, and piecewise

Double-check your arithmetic — these problems reward careful calculation