AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-7)
Unit 2 · Topic 2.7
2.7 Composition of Functions
Composing functions means feeding the output of one function into another: f(g(x)) runs g first, then f. You'll compute compositions from formulas, tables and graphs, find their domains, and break complicated functions into simpler pieces.
Key terms
- composition
- composite function
- inner and outer function
- identity function
What f(g(x)) means
The composite function f ∘ g, read “f composed with g,” sends an input x through g first and then sends that output through f: (f ∘ g)(x) = f(g(x)). g is the inner function and f is the outer function.
Work from the inside out. To find f(g(2)), first find g(2), then plug that number into f.
Order matters
Composition is not commutative: f(g(x)) and g(f(x)) are usually different. Putting on socks and then shoes is not the same as shoes and then socks.
Composition is also different from multiplication. f(g(x)) is not f(x) · g(x).
Formulas, tables and graphs
With formulas: replace every x in f with the whole expression g(x). If f(x) = x² − 1 and g(x) = 2x + 3, then f(g(x)) = (2x + 3)² − 1.
With tables or graphs: read the value of g, then look up that value as an input to f. A table of f ∘ g can be built point by point: for each x, record (x, f(g(x))).
The domain of a composition
x must be in the domain of g, and g(x) must be in the domain of f. So the domain of f ∘ g contains only the inputs of g whose outputs f can accept.
Example: f(x) = √x and g(x) = x − 5. f(g(x)) = √(x − 5) needs x − 5 ≥ 0, so the domain is x ≥ 5.
The identity function and decomposition
The identity function I(x) = x leaves every input unchanged, so f(I(x)) = I(f(x)) = f(x). It plays the role that 0 plays in addition and 1 plays in multiplication.
Decomposition runs the other way: write a complicated function as a composition of simpler ones. h(x) = (3x − 1)⁴ is f(g(x)) with g(x) = 3x − 1 and f(x) = x⁴. Done properly, the inner function's expression replaces every x in the outer function.
Transformations are compositions too. f(x) + k is f followed by “add k,” and f(x + k) is “add k” followed by f. Likewise k · f(x) and f(kx) compose f with “multiply by k.” Composition also links two quantities without a direct formula: if area depends on radius and radius depends on time, then area as a function of time is a composition.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Composing formulas
Let f(x) = x² − 1 and g(x) = 2x + 3. Find f(g(x)) and g(f(x)).
Show the solutionHide the solution
- Step 1: f(g(x)) = (2x + 3)² − 1 = 4x² + 12x + 9 − 1 = 4x² + 12x + 8.
- Step 2: g(f(x)) = 2(x² − 1) + 3 = 2x² − 2 + 3 = 2x² + 1.
- Step 3: The two results are different, showing that order matters.
Answer: f(g(x)) = 4x² + 12x + 8; g(f(x)) = 2x² + 1.
- Example 2
Composing from a table
The table gives f and g at x = 0, 1, 2, 3, 4: f values 3, 4, 0, 2, 1 and g values 2, 0, 4, 1, 3, in that order. Find f(g(2)), g(f(2)) and f(f(0)).
Show the solutionHide the solution
- Step 1: f(g(2)): g(2) = 4, then f(4) = 1.
- Step 2: g(f(2)): f(2) = 0, then g(0) = 2.
- Step 3: f(f(0)): f(0) = 3, then f(3) = 2.
Answer: f(g(2)) = 1, g(f(2)) = 2, f(f(0)) = 2.
- Example 3
Trap: domain of a composition
Let f(x) = √x and g(x) = x − 5. Find the domain of f(g(x)), and evaluate f(g(3)) if possible.
Show the solutionHide the solution
- Step 1: f(g(x)) = √(x − 5). The square root needs x − 5 ≥ 0, so x ≥ 5.
- Step 2: g(3) = −2 is a perfectly good output of g, but f can't take −2 as an input (no real square root).
- Step 3: So 3 is not in the domain of f ∘ g, even though it's in the domain of g.
Answer: The domain is x ≥ 5; f(g(3)) is undefined.
Common mistakes
- Working from the outside in. In f(g(x)), g acts first.
- Treating f(g(x)) as the product f(x) · g(x).
- Replacing only some of the x's when substituting. Every x in the outer function gets the whole inner expression, in parentheses.
- Ignoring the domain: a composition can be undefined where the inner function is fine.
On the exam
- Composition from tables and graphs shows up often in multiple choice and in the function-concepts free-response question. Write each step, such as “g(2) = 4, so f(g(2)) = f(4) = 1.”
- Composition is the key to inverses (topic 2.8): f(f⁻¹(x)) = x.
Connected topics
Videos
Check yourself
4 questions on 2.7 Composition of Functions. Pick an answer to see if you got it, and why.
Let h(x) = (2x − 1)³ + 4. If h(x) = f(g(x)) and g(x) = 2x − 1, which of the following could be f(x)?
Let f(x) = x² − 1 and g(x) = 2x + 3. Which of the following is f(g(x))?
The temperature of an oven, in degrees Celsius, t minutes after it is turned on is C(t). The function F(c) = 1.8c + 32 converts a temperature of c degrees Celsius to degrees Fahrenheit. Which of the following gives the temperature of the oven, in degrees Fahrenheit, 4 minutes after it is turned on?
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| f(x) | 2 | 4 | 1 | 0 | 3 |
| g(x) | 3 | 0 | 4 | 1 | 2 |
Table of values
What is the value of f(g(2))?
0 of 4 answered