For f(x) = 2x + 1 and g(x) = x², find f(g(3)).
- First g(3) = 9.
- Then f(9) = 2(9) + 1.
Answer: 19
Composition of Functions Worksheets
Evaluate and transform functions while keeping the input order and domain explicit.
Learn the skill
Composing functions in the correct order
The composition (f ∘ g)(x) means f(g(x)): apply g first, then feed its output into f. Both steps must be defined. Reversing the order generally produces a different function.
A method you can use
- Read from the inside outward to identify which function acts first.
- Substitute the entire inner expression for each input in the outer function.
- Check that the inner input is allowed and its output lies in the outer function’s domain.
Worked examples
For the same functions, find g(f(3)).
- First f(3) = 7.
- Then g(7) = 7².
Answer: 49
For f(x) = 1/x and g(x) = x − 2, find f(g(x)) and its domain.
- Replace the input to f with x − 2: 1/(x − 2).
- The denominator cannot be zero, so x cannot be 2.
Answer: 1/(x − 2), with x ≠ 2
Try it yourself
Solve these before opening the answers. The printable sets below give you more practice.
1. For f(x) = x + 4 and g(x) = 3x, find f(g(2)).
Show answer and explanation
10
g(2) = 6, then f(6) = 10.
2. For f(x) = x² and g(x) = x − 1, find g(f(3)).
Show answer and explanation
8
f(3) = 9, then g(9) = 8.
Before you begin
Evaluate functions, solve for variables and use exponential/logarithmic inverses.
Choose your level
These sets use the same exercise types and number ranges across level labels. Choose a fresh set for additional practice.
- Easy: Start with the preview and follow the worked example.
- Medium: Practice the same exercise types with a fresh numbered set.
- Hard: Use another set to practice explaining and checking your answers.
Questions and answer calculations checked on 2026-09-17. How we check worksheets. Free for personal and classroom use.
Preview and print an easy set
Try six questions below, or choose a full set at your level.
1
Let f(x) = 3x + 4 and g(x) = x² + 5. Find f(g(2)).
2
Let f(x) = 2x + 4 and g(x) = x² + 7. Find f(g(1)).
3
Let f(x) = 2x + 6 and g(x) = x² + 3. Find f(g(-1)).
4
Let f(x) = 6x + 8 and g(x) = x² + 2. Find f(g(5)).
5
Let f(x) = 8x + 3 and g(x) = x² + 6. Find f(g(0)).
6
Let f(x) = 3x + 1 and g(x) = x² + 1. Find f(g(-2)).
Answer Key
1. 312. 203. 144. 1705. 516. 16