AP Precalculus · AP Precalculus CED Unit 2 · 14 min read
1. Introduction to Composition of Functions★★☆☆☆⏱ 3 min
Composition of functions is the process of using the output of one function as the input of a second function, combining multiple functions into a single new function. Per AP Precalculus CED, this is a core skill in Unit 2, appearing in 8-12% of Unit 2 exam questions on both multiple-choice and free-response sections.
In AP Precalculus Unit 2, composition most frequently pairs exponentials or logarithms with polynomials, or pairs exponential and logarithmic functions with each other, especially when working with inverse functions.
2. Evaluating Composite Functions★★☆☆☆⏱ 3 min
Evaluating a composite function for a given input $x$ follows one core rule: always work from the inside out. First evaluate the inner function (the function closest to $x$ in notation), then substitute that output into the outer function as the new input.
Exam tip: Always explicitly confirm the order of composition before starting calculations to avoid falling for common distractors.
3. Finding Composite Rules and Domains★★★☆☆⏱ 4 min
To find the general algebraic rule for $(f \circ g)(x)$, substitute the entire expression for $g(x)$ into the outer function $f$ in place of $x$. A frequently tested skill is finding the domain of the new composite function, which must satisfy two conditions: 1) $x$ is in the domain of the inner function $g$, and 2) the output $g(x)$ is in the domain of the outer function $f$.
4. Composition of a Function and Its Inverse★★★☆☆⏱ 4 min
A key property of inverse functions is that composing a function with its inverse gives the identity function, which outputs the original input. For any one-to-one function $f$ with inverse $f^{-1}$, two core identities hold:
(f \circ f^{-1})(x) = f(f^{-1}(x)) = x
(f^{-1} \circ f)(x) = f^{-1}(f(x)) = x
For Unit 2, this property is most commonly applied to inverse pairs of exponential and logarithmic functions: for $b>0, b \neq 1$, $f(x) = b^x$ and $f^{-1}(x) = \log_b x$. This gives the identities $b^{\log_b x} = x$ (for $x>0$) and $\log_b(b^x) = x$ (for all real $x$), which are used constantly to simplify expressions and solve equations.
5. AP-Style Concept Check★★★★☆⏱ 4 min
Common Pitfalls
Why: Students confuse the order of notation, forgetting the function closest to $x$ is the inner function evaluated first.
Why: Simplification can cancel terms that introduced domain restrictions, leading to incorrectly including disallowed inputs.
Why: Students memorize the identity without remembering the domain restriction on the logarithm.
Why: Students confuse composition with multiplication of functions, which is commutative.
Why: Students reverse the power rule $\ln x^a = a \ln x$, misapplying the coefficient as a multiplicative factor instead of an exponent.
Why: Students only check the outer function's restrictions, forgetting the inner function already has a domain restriction.