15.1 Functions: inverse and composite — Learn

Prerequisites: substitute values into expressions and solve one-step linear equations.

Objectives: find inverse functions, evaluate composite functions, and state where a rule is defined.

A function is a rule that gives one output for each allowed input. The notation $f(x)$ means the output of function $f$ for input $x$. An inverse function reverses the rule. A composite function applies one rule and then another: $g(f(x))$ means apply $f$ first.

Example 1. Find the inverse of $f(x)=3x+5$.

Write $y=3x+5$. Swap input and output: $x=3y+5$. Subtract $5$: $x-5=3y$. Divide by $3$: $y=\frac{x-5}{3}$. Therefore $f^{-1}(x)=\frac{x-5}{3}$.

Example 2. Let $f(x)=2x+1$ and $g(x)=x^2$. Find $g(f(x))$ and $f(g(x))$.

$g(f(x))=(2x+1)^2=4x^2+4x+1$. $f(g(x))=2x^2+1$. The order matters because these expressions differ.

A denominator cannot be zero. For $r(x)=\frac{1}{x-4}$, the input $4$ is excluded.

Watch out. In $g(f(x))$, apply $f$ first. At $x=1$, $g(f(1))=9$ while $f(g(1))=3$. The results differ, so order matters.

Predict: before moving the sliders, predict the inverse of $f(x)=2x+3$. Write your prediction down first.

Test: adjust the sliders to test different rules and compare each inverse with your prediction.

Explain: why does undoing a rule require reversing its operations? Write your explanation before reading on.

Progress check

For the chapter machines, the second rule reverses the first. Check this by composing their expressions in both orders.

You should now be able to

• find an inverse function by swapping input and output;
• evaluate a composite function in the stated order;
• identify inputs that make a denominator zero.