Study guide: Precalculus: Composition and Inverse Functions
Function operations must respect input order and domain. A composition feeds one output into another function, while an inverse reverses a one-to-one relationship and exchanges the original domain and range.
Ideas to explain in your own words
- In f(g(x)), g acts on the supplied input before f acts on its result. This nested process generally differs from g(f(x)) even when both functions are simple. Read the innermost operation first.
- A composed function exists only when its inner function is defined and that output belongs to the outer function’s domain. A square-root outer function requires its incoming quantity to be nonnegative. Checking the inner function’s domain alone is insufficient.
- A nonconstant linear function is one-to-one. Solving its output equation for the original input undoes its addition and multiplication in reverse order, producing the inverse function. An inverse-function symbol is not a reciprocal exponent.
- Squaring on all real inputs loses sign information and is not one-to-one. Restricting to nonnegative inputs makes the positive square root recover one unique input and supplies an inverse. Do not return both square-root branches as a function output.
Useful vocabulary
- composition
- Applying a function to another function’s output.
- one-to-one
- Different inputs have different outputs.
- inverse function
- A function reversing another function’s input-output relationship.
Watch your reasoning
- Read the innermost operation first. Composition is not multiplication of function values; record the intermediate output so the outer function receives the computed value rather than the original input.
- Checking the inner function’s domain alone is insufficient. Translate the outer restriction into an inequality on the original input, and keep an inclusive endpoint when a zero radicand is allowed.
- An inverse-function symbol is not a reciprocal exponent. Check by composing in both orders; a valid inverse returns the starting input, subject to the functions’ stated domains.
A worked review example
Complete supplied guide examples: Follow the Inner Function before the Outer One — Let f(x)=x² and g(x)=2x+1. Find (f∘g)(2). Candidate answers: (A) 24 (B) 25 (C) 26 (D) 50. Composition applies g first: g(2)=2(2)+1=5. Then apply f: (5)²=25. Reversing the order would give g(f(2)) instead. Require the Inner Output to Fit the Outer Domain — Let f(u)=√u and g(x)=x−2. What is the real domain of f(g(x))? Candidate answers: (A) Every real x (B) x≠2 (C) x≤2 (D) x≥2. The output of g must be in the square-root domain, so x−2≥0. Thus x≥2, including the endpoint where the root is zero. Reverse a Linear Function’s Operations — Find the inverse of f(x)=3x+2. Candidate answers: (A) f⁻¹(x)=(x+2)/3 (B) f⁻¹(x)=(x−2)/3 (C) f⁻¹(x)=1/(3x+2) (D) f⁻¹(x)=3x−2. Set y=3x+2, then solve x=(y−2)/3. Swap the input/output labels. An inverse undoes a function; it is not generally its reciprocal. Restrict a Quadratic to Make an Inverse Function — The function f(x)=x²+2 has domain x≥0. Find its inverse and domain. Candidate answers: (A) f⁻¹(x)=x²−2, x≥0 (B) f⁻¹(x)=±√(x−2), all real x (C) f⁻¹(x)=√(x+2), x≥0 (D) f⁻¹(x)=√(x−2), x≥2. The nonnegative domain makes f one-to-one. Solve y=x²+2 with x≥0: x=√(y−2). The range of f, [2,∞), becomes the inverse domain.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.