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Function fundamentals
Composite and inverse functions
CatalogueDPMathematicsMathematics: Analysis and Approaches HLFunctions

Composite and inverse functions

Compose functions with valid domains and find inverses only after checking one-to-one behavior.

  • AA HL
  • Function fundamentals
  • Papers 1 and 2
  • No GDC needed
  • About 20 min

By the end of this lesson

  • evaluate and form composite functions
  • track restrictions through a composition
  • find and verify an inverse on a suitable domain

Why this matters. An algebraic rule is not a complete function without its domain. Composition and inversion both depend on which inputs are actually allowed.

Composition follows an order

In $(f\circ g)(x)=f(g(x))$, apply $g$ first. The output of $g$ must lie in the domain of $f$.

Checkpoint

Let $f(x)=\sqrt{x}$ and $g(x)=x-3$. What restriction applies to $(f\circ g)(x)$?
Show explanation

The input to $f$ is $x-3$, so require $x-3\ge0$.

Example 1

Build a composition

Keep the inner function visible

Let $f(x)=\dfrac1{x-2}$ and $g(x)=3x+1$. Find $f(g(x))$ and its domain.

$$f(g(x))=\frac1{(3x+1)-2}=\frac1{3x-1}.$$

The denominator cannot be zero, so $x\ne\frac13$.

Why not just use $x\ne2$?

The input reaching $f$ is $g(x)$, so require $g(x)\ne2$. This gives $3x+1\ne2$.

Check one-to-one before inverting

An inverse reverses a function. It is a function only when each output of the original rule comes from one allowed input.

Checkpoint

Can $f(x)=x^2$ with domain $x\in\mathbb R$ have an inverse function?
Show explanation

Both $a$ and $-a$ give the same output. Restricting to $x\ge0$ or $x\le0$ makes the rule one-to-one.

Example 2

Find an inverse

Reverse each operation

Find the inverse of $f(x)=\dfrac{2x-5}{3}$.

Write $y=f(x)$
$$y=\frac{2x-5}{3}$$
Solve for $x$
$$3y=2x-5\Rightarrow x=\frac{3y+5}{2}$$
Rename the input
$$f^{-1}(x)=\frac{3x+5}{2}.$$

A check gives $f(f^{-1}(x))=x$.

Order the method

Put these inverse steps in a sensible order.

  1. Write $y=f(x)$ and interchange the roles of input and output
  2. Solve the resulting equation for $y$
  3. State the inverse domain and range
  4. Check a composition gives $x$
Show a valid order
  1. Write $y=f(x)$ and interchange the roles of input and output
  2. Solve the resulting equation for $y$
  3. State the inverse domain and range
  4. Check a composition gives $x$

The algebra, restrictions and verification now follow logically.

Example 3

Restrict, then invert

A quadratic branch

Let $h(x)=(x-1)^2+4$ for $x\ge1$. Find $h^{-1}$.

$$y=(x-1)^2+4\Rightarrow x-1=\sqrt{y-4}$$

The positive root is required because $x\ge1$.

$$h^{-1}(x)=1+\sqrt{x-4},\qquad x\ge4.$$

The inverse graph is the reflection of the restricted branch in $y=x$.

In the exam

Quick check

Five questions to check the main decisions from this lesson.

Your score is not saved.

  1. Which function acts first in $f(g(x))$?
    Show explanation

    $g$ produces the input used by $f$.

  2. If $f(x)=2x+1$, what is $f^{-1}(x)$?
    Show explanation

    Reverse adding $1$, then multiplying by $2$.

  3. Why does unrestricted $x^2$ have no inverse function?
    Show explanation

    Two different inputs can give the same output.

  4. The graph of $f^{-1}$ is the reflection of $f$ in…
    Show explanation

    Inversion exchanges coordinates $(x,y)$ and $(y,x)$.

  5. For $f(x)=1/x$, what input is excluded from $f(f(x))$?
    Show explanation

    Both occurrences of $1/x$ require a non-zero input.

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