Functions, composition and inverse inputs

A function assigns one output to each allowed input. Composition feeds one function output into another, and an inverse reverses an input-output relationship when that reversal is itself a function.

A reliable method

Work from the inside outward in f(g(x)). For a linear inverse calculation, set the output equal to y, solve for the original input, and check by composing back.

The common trap

The product f(x)g(x) is not the composition f(g(x)). Parentheses describe an input operation; a multiplication symbol describes a different calculation.

Check and transfer

A formula can impose restrictions before evaluation: an even-root radicand must be nonnegative over the reals, and a denominator must not vanish. Include those conditions when deciding the domain.

Worked comparison: identify the structure

If f(x)=2x+3 and g(x)=x²−1, then f(g(2))=f(3)=9. Reversing order gives g(f(2))=g(7)=48, while multiplication gives f(2)g(2)=21. Three valid operations produce different answers. Read the notation first.

Transfer and explain

For f(x)=3x−4, solve 3x−4=8 to get f⁻¹(8)=4. Check f(4)=8. A reciprocal 1/f(8)=1/20 is a different operation entirely.

WORKED EXAMPLE

The two function rules are shown in the table. Find f(g(2)).

Table: f(t)=2t+1; g(t)=t squared minus 3.

First evaluate g at the given input.
Feed that output into f; do not multiply f and g.
The requested value is 3. Substitute or reverse the operation to check this result.

Answer: 3

Compare the other answer choices

Multiply f(x) and g(x) instead of composing.

Double x squared but not the entire inner output x squared minus three.

Reverse the order and calculate g(f(x)).

TAKE IT FURTHER · WRITTEN REASONING

Explain your thinking.

Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.

Explain the structure you use, show the intermediate calculation and refute one tempting alternative. Self-assessment only; no automatic points.

Your reasoning

This draft stays on this page only. Use paper if you want to keep your working.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

CLEP College Algebra exam overview Siyavula Grade 10 foundation reference

Functions, composition and inverse inputs · CLEP College Algebra