Inverse functions are a vital concept in A-Level Maths, offering insights into the relationship between inputs and outputs of a function. They are essential for understanding mathematical operations from a different perspective.

**Understanding Inverse Functions**

**Definition**

An inverse function, denoted as $f^{-1}(x)$, reverses the action of a function. For example, if $f(3) = 5$, then $f^{-1}(5) = 3$. The graphs of $y = f(x)$ and $y = f^{-1}(x)$ are symmetrical about the line $y = x$.

**Properties**

A fundamental property of inverse functions is:

$f(f^{-1}(x)) = f^{-1}(f(x)) = x$

This reflects that applying a function followed by its inverse returns the original value.

**Criteria for the Existence of Inverse Functions**

For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

**Steps to Find an Inverse Function**

**Procedure**

1. **Verify One-to-One Nature:** Ensure the function is one-to-one.

2. **Rearrange as ( y ):** Express $f(x)$ as $y$.

3.** Solve for ( x ):** Rearrange to make $x$ the subject.

4.** Swap Variables:** Replace each $x$ with $y$.

5. **Represent Inverse Function:** Substitute $y$ with $f^{-1}(x)$.

**Example 1**

Given $f(x) = 3x + 4$:

1. Write as $y$: $y = 3x + 4$.

2. Rearrange for $x$: $x = \frac{y - 4}{3}$.

3. Swap $x$ and $y$: $y = \frac{x - 4}{3}.$

4. Inverse notation: $f^{-1}(x) = \frac{x - 4}{3}$.

**Example 2**

For $f(x) = 2x - 5$:

1. Express as $y$: $y = 2x - 5.$

2. Solve for $x$: $x = \frac{y + 5}{2}$.

3. Interchange $x$ and $y$: $y = \frac{x + 5}{2}$.

4. Represent inverse: $f^{-1}(x) = \frac{x + 5}{2}$.

**Example 3**

Consider $f(x) = \frac{1}{x}$:

1. Set as $y$: $y = \frac{1}{x}$.

2. Rearrange for $x$: $x = \frac{1}{y}$.

3. Replace $x$ with $y$: $y = \frac{1}{x}$.

4. Inverse form: $f^{-1}(x) = \frac{1}{x}$.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.