Composition and Inverse Functions
A free College Algebra lesson from the “Functions” unit, with a worked example and practice problems including step-by-step solutions.
Composition uses one function's output as another function's input. Inverse functions reverse the input-output relationship of the original function.
What you'll learn
- Evaluate composite functions
- Find simple inverse functions
- Interpret inverse relationships
Worked example
Problem. If f(x) = 2x + 1 and g(x) = x - 4, find f(g(9)).
- Evaluate inside first: g(9) = 5.
- Now evaluate f(5).
- 2(5) + 1 = 11.
Answer: 11
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. If f(x) = x + 2 and g(x) = 3x, find f(g(4)).
Show solution
- g(4) = 12, then f(12) = 14.
Answer: 14
2. The inverse of f(x) = x - 8 is...
Choices: f^-1(x) = x + 8 · f^-1(x) = x - 8 · f^-1(x) = 8x · f^-1(x) = x/8
Show solution
- Undo subtracting 8.
Answer: f^-1(x) = x + 8
3. The inverse of f(x) = 4x + 1 is...
Choices: f^-1(x) = (x - 1)/4 · f^-1(x) = (x + 1)/4 · f^-1(x) = 4x - 1 · f^-1(x) = x/4 + 1
Show solution
- Swap x and y, then solve.
Answer: f^-1(x) = (x - 1)/4
4. If f(x) = x - 5 and g(x) = 2x + 1, find f(g(6)).
Show solution
- g(6) = 13, then f(13) = 8.
Answer: 8
5. The inverse of f(x) = 2x - 7 is...
Choices: f^-1(x) = (x + 7)/2 · f^-1(x) = 2x + 7 · f^-1(x) = (x - 7)/2 · f^-1(x) = x/2 - 7
Show solution
- Undo subtracting 7, then undo multiplying by 2.
Answer: f^-1(x) = (x + 7)/2
Practice this interactively with instant feedback and an AI tutor.
Practice Composition and Inverse Functions Take the free placement check