Evaluating Functions and Composition
Composition $f(g(x))$ means run one function, then feed the result into the next. Work from the inside out.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
To find $f(g(x))$, evaluate $g$ first, then put that output into $f$. It is not $f(x) \cdot g(x)$.
How to solve one, step by step
Example: $f(x) = 2x + 1$, $g(x) = x^2$. Find $f(g(3))$.
- Inside first: $g(3) = 9$.
- Then $f(9) = 2(9) + 1 = 19$.
The mistakes that cost points
- Multiplying instead of composing. $f(g(x))$ is a substitution, not a product.
- Replacing only some of the variable. Substitute the inner function for every $x$ in the outer function.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
Let $f(x) = 2x + 1$ and $g(x) = x - 3$. What is $f(g(5))$?
- A$12$
- B$5$
- C$3$
- D$7$
Show solution
Answer: B, $5$. Step 1: Evaluate the inner function first. $g(5) = 5 - 3 = 2$. Step 2: Use that result as the input to $f$. $f(2) = 2(2) + 1 = 5$.
Medium
Let $f(x) = 3x - 2$ and $g(x) = x^2 + 1$. What is $f(g(2))$?
- A$7$
- B$10$
- C$13$
- D$25$
Show solution
Answer: C, $13$. Step 1: Evaluate $g(2) = 2^2 + 1 = 5$. Step 2: Evaluate $f(5) = 3(5) - 2 = 13$.
Hard
Let $f(x) = \sqrt{x + 1}$ and $g(x) = x^2 - 5$. What is $f(g(3))$?
- A$3$
- B$2\sqrt{5}$
- C$\sqrt{10}$
- D$\sqrt{5}$
Show solution
Answer: D, $\sqrt{5}$. Step 1: Evaluate $g(3) = 3^2 - 5 = 9 - 5 = 4$. Step 2: Evaluate $f(4) = \sqrt{4 + 1} = \sqrt{5}$.
Find your exact gaps
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