AP PreCalculus Flashcards: Composition of Functions
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 14 cards to help you master important concepts.
Describe the flow of input and output values in the composite function g ∘ f.
An input value `x` first goes into function `f`. The resulting output, `f(x)`, is then used as the input for function `g` to produce the final output `g(f(x))`.
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Describe the flow of input and output values in the composite function g ∘ f.
An input value `x` first goes into function `f`. The resulting output, `f(x)`, is then used as the input for function `g` to produce the final output `g(f(x))`.
Rewrite the function h(x) = (2x - 1)³ as a composition of two functions, f(x) and g(x).
One possible decomposition is to let the inner function be g(x) = 2x - 1 and the outer function be f(x) = x³.
Given f(x) = 3x + 2 and g(x) = x², evaluate the composition (f ∘ g)(3).
First, evaluate the inner function: g(3) = 3² = 9. Then, use this output as the input for the outer function: f(9) = 3(9) + 2 = 29.
Let f(x) = 1/x and g(x) = x + 5. Evaluate (g ∘ f)(2).
First, evaluate the inner function: f(2) = 1/2. Then, use this result as the input for the outer function: g(1/2) = 1/2 + 5 = 5.5.
Decompose the function h(x) = cos(x²).
This function can be decomposed by letting the inner function be g(x) = x² and the outer function be f(x) = cos(x).
Is the composition of functions commutative? Explain what this means for f ∘ g and g ∘ f.
Function composition is not commutative. This means that f ∘ g and g ∘ f are typically different functions and will produce different outputs.
What is the identity function in the context of function composition?
The identity function is f(x) = x. When composed with any function g, the resulting composite function is the same as g.
What is the composite function f ∘ g?
The composite function f ∘ g maps a set of input values to a set of output values such that the output values of g are used as input values of f.
What does it mean to decompose a function?
Decomposing a function means to rewrite a given function as a composition of two or more less complicated functions.
Given f(x) = √x and g(x) = x - 4, construct a representation of the composite function (f ∘ g)(x).
To find f(g(x)), substitute the entire expression for g(x) into f(x). This results in the composite function f(g(x)) = √(x - 4).
How is an analytic representation of f(g(x)) constructed?
An analytic representation of f(g(x)) is constructed by substituting the expression for g(x) for every instance of x in the function f.
Using f(x) = x + 5 and g(x) = 2x, show that f(g(x)) ≠ g(f(x)).
f(g(x)) = f(2x) = 2x + 5. In contrast, g(f(x)) = g(x + 5) = 2(x + 5) = 2x + 10. Since 2x + 5 ≠ 2x + 10, the composition is not commutative.
What is the key difference between evaluating (f ∘ g)(x) and (g ∘ f)(x)?
The key difference is the order of operations. For (f ∘ g)(x), you apply g first, then f; for (g ∘ f)(x), you apply f first, then g.
If f(x) = 5x and g(x) = x² - 2, construct the analytic representation for g(f(x)).
Substitute the expression for f(x) into g(x): g(f(x)) = g(5x) = (5x)² - 2 = 25x² - 2.