# Inverse functions: algebraic solution

> IB Mathematics: Applications and Interpretation HL · Unit 2: Functions
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u2-inverse-functions-algebraic-solution/

This module covers how to find inverses of one-to-one functions using algebraic methods, verify inverse pairs via composition, and adjust domain and range to meet inverse function requirements for IB AI HL exams.

**Prerequisites:** [Function notation and composition](https://www.owlsprep.com/study/ib-math-ai-hl-u2-function-composition/); [Domain and range of functions](https://www.owlsprep.com/study/ib-math-ai-hl-u2-domain-range/)

## Learning objectives

- Define inverse functions and identify their core properties
- Apply the step-by-step algebraic method to find inverses of one-to-one functions
- Verify inverse function pairs using function composition
- Correctly state domain and range for inverse functions

## Key Definitions and Verification of Inverses

**Inverse Function** — For a one-to-one function $f$ with domain $D$ and range $R$, the inverse function $f^{-1}$ has domain $R$ and range $D$, and satisfies $f(f^{-1}(x)) = x$ for all $x \in R$ and $f^{-1}(f(x)) = x$ for all $x \in D$.

*Notation:* $f^{-1}(x)$

*Example:* If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$

Only one-to-one functions (functions that pass the horizontal line test) have inverses. If a function is not one-to-one over its entire domain, we restrict the domain to make it one-to-one before calculating the inverse.

**Worked example:** Verify that $f(x) = 4x - 5$ and $g(x) = \frac{x + 5}{4}$ are inverses of each other.

1. First check the composition $f(g(x))$ by substituting $g(x)$ into $f$:

   $$f(g(x)) = f\left(\frac{x+5}{4}\right) = 4\left(\frac{x+5}{4}\right) - 5 = (x+5) - 5 = x$$
2. Next check the reverse composition $g(f(x))$ by substituting $f(x)$ into $g$:

   $$g(f(x)) = g(4x - 5) = \frac{(4x - 5) + 5}{4} = \frac{4x}{4} = x$$
3. Since both compositions simplify to $x$, the functions meet the definition of inverses.

*Calculator:* forbidden

## Step-by-Step Algebraic Method for Finding Inverses

1. Start with $y = f(x)$, note the original domain and range
2. Swap the positions of $x$ and $y$ (this reflects over $y=x$)
3. Rearrange the equation to solve for $y$ in terms of $x$
4. Replace $y$ with $f^{-1}(x)$
5. State the domain of $f^{-1}(x)$, which equals the range of the original $f(x)$

**Worked example:** Find the inverse of $f(x) = 2\sqrt{x + 3}$ for $x \geq -3$.

1. Set $y$ equal to $f(x)$:

   $$y = 2\sqrt{x + 3}, \quad x \geq -3$$
2. Swap $x$ and $y$ to get the inverse relationship:

   $$x = 2\sqrt{y + 3}$$
3. Solve for $y$: divide by 2 then square both sides:

   $$\frac{x}{2} = \sqrt{y + 3} \implies \left(\frac{x}{2}\right)^2 = y + 3$$
4. Isolate $y$:

   $$y = \frac{x^2}{4} - 3$$
5. The original function $f(x)$ has range $y \geq 0$, so the inverse has domain $x \geq 0$:

   $$f^{-1}(x) = \frac{x^2}{4} - 3, \quad x \geq 0$$

> **Exam tip:** Examiners almost always award a separate mark for correctly stating the domain of the inverse function — never skip this step.

*Calculator:* forbidden

## Inverses of Restricted Quadratic Functions

Quadratic functions are not one-to-one over their full domain, so we always restrict the original domain to one side of the vertex to make it invertible. The original domain restriction tells us which root to pick when solving for the inverse.

**Worked example:** Find the inverse of $f(x) = (x - 2)^2 - 4$ for $x \geq 2$.

1. Set $y$ equal to $f(x)$:

   $$y = (x - 2)^2 - 4, \quad x \geq 2$$
2. Swap $x$ and $y$:

   $$x = (y - 2)^2 - 4$$
3. Rearrange to isolate the squared term:

   $$x + 4 = (y - 2)^2$$
4. Take the square root of both sides, keeping the absolute value:

   $$\sqrt{x + 4} = |y - 2|$$
5. Original domain $x \geq 2$ means inverse range is $y \geq 2$, so $y-2$ is non-negative:

   $$\sqrt{x + 4} = y - 2$$
6. Original range of $f(x)$ is $y \geq -4$, so inverse domain is $x \geq -4$:

   $$f^{-1}(x) = \sqrt{x + 4} + 2, \quad x \geq -4$$

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Forgetting to state the domain of the inverse function
  - Why it fails: Examiners require the domain to confirm you understand that inverse domain equals original range, skipping this loses easy marks
  - Correct: Always write the domain of $f^{-1}(x)$ after finding the algebraic expression
- **Wrong:** Interpreting $f^{-1}(x)$ as $\frac{1}{f(x)}$
  - Why it fails: The $-1$ exponent in inverse notation does not mean reciprocal — this is a common beginner mistake
  - Correct: Remember $f^{-1}(x)$ undoes the action of $f(x)$, it is not the reciprocal
- **Wrong:** Keeping both positive and negative roots when solving for inverse quadratics
  - Why it fails: A function can only have one output per input, so two roots would not give a valid function
  - Correct: Use the original domain restriction to select only one valid root for the inverse
- **Wrong:** Swapping variables after solving for $y$ instead of before
  - Why it fails: This leaves you with the inverse expressed in terms of the wrong variables, giving an incorrect final result
  - Correct: Always swap $x$ and $y$ immediately after writing $y = f(x)$

## Cheatsheet

| Step | Action | Exam Note |
| --- | --- | --- |
| 1 | Start with $y = f(x)$ | Write original domain/range |
| 2 | Swap $x$ and $y$ | Reflects over $y=x$ |
| 3 | Solve for $y$ | Pick one root for quadratics |
| 4 | Write $f^{-1}(x) = y$ | Set domain = original range |
| 5 | Verify (optional) | Check $f(f^{-1}(x)) = x$ to confirm |

## What's next

Now that you can find inverses algebraically, you can extend this understanding to graphical representations of inverse functions and core applications like exponential and logarithmic inverse pairs, which are heavily tested in IB AI HL. Inverse functions are foundational for many topics in Unit 2 and beyond, including solving equations involving composite functions and modelling inverse relationships in real-world contexts. Mastering algebraic inversion will also help you quickly recognize inverse pairs on exam day, saving you valuable time on both Paper 1 and Paper 2.

- [Modelling with functions](https://www.owlsprep.com/study/ib-math-ai-hl-u2-modelling-with-functions/)
- [Geometry and trigonometry](https://www.owlsprep.com/study/ib-math-ai-hl-u3-overview/)
- [2D and 3D coordinate geometry](https://www.owlsprep.com/study/ib-math-ai-hl-u3-2d-and-3d-coordinate-geometry/)

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