In this section, we explore the graphical interpretation of inverse functions, a crucial aspect of A-Level Pure Mathematics. Understanding how to find and interpret the inverses of functions graphically, particularly focusing on reflection over the line $y = x$, is essential. We will also delve into techniques for sketching functions and their inverses on the same axes, demonstrating symmetry with respect to the line $y = x$.

**Understanding Inverse Functions**

Inverse functions are a key concept in mathematics, providing a way to 'undo' the effect of a function. They are defined such that if $f(x) = y$, then $f^{-1}(y) = x$. This relationship means that the inverse function reverses the action of the original function.

**Characteristics of Inverse Functions**

**Reflection**: Inverse functions are a reflection of the original function across the line $y = x$. If a point $(a, b)$ lies on the graph of the function, then the point $(b, a)$ will lie on the graph of its inverse.**Domain and Range**: The domain of the original function becomes the range of its inverse, and vice versa.**One-to-One Requirement**: For a function to have an inverse, it must be one-to-one (injective). This means that each output is produced by exactly one input.

**Sketching Inverse Functions**

Sketching the inverse of a function graphically involves a few steps:

**Plot the Original Function**: Begin by plotting the original function on a set of axes.**Reflect Across**$y = x$: Reflect each point of the original function across the line $y = x$. This can be done by swapping the x and y coordinates of each point.**Plot the Inverse Function**: The reflected points form the graph of the inverse function.

**Symmetry with the Line **$y = x$

The symmetry of a function and its inverse about the line $y = x$ is a visual representation of their inverse relationship. This symmetry can be used as a check to ensure the accuracy of the sketched inverse.

**Transformations of Graphs**

Understanding how transformations affect the graph of a function and its inverse is crucial. These transformations include translations, stretches, and reflections.

**Vertical and Horizontal Translations**

When a graph is translated vertically or horizontally, the symmetry with the line $y = x$ is maintained, but the specific points of reflection shift. For example, translating $f(x)$ up by 2 units will also shift $f^{-1}(x)$ right by 2 units.

**Stretches and Reflections**

Stretches and reflections alter the shape and position of the graph but do not disrupt the symmetry about $y = x$. For instance, reflecting $f(x)$ across the x-axis will reflect $f^{-1}(x)$ across the y-axis.

**Combining Transformations**

When applying multiple transformations to a function, the inverse undergoes corresponding transformations. The symmetry about $y = x$ remains consistent, serving as a guide to understanding these changes.

**Examples**

**Example 1: **

Finding and Sketching the Inverse of $f(x) = 2x + 3.$

**Solution:**

**1. Draw Axes**: Make two lines crossing at the middle for the x-axis and y-axis.

**2. Plot **$f(x) = 2x + 3$:

- Mark points for $x$ values (like 0 and 1) on the graph.
- Connect them with a line going up to the right.

**3. Plot **$f^{-1}(x) = \frac{x - 3}{2}$:

- Mark points for $x$ values (like 3 and 5).
- Connect them with a line going up, but less steep than the first.

**4. Check Symmetry**:

- The lines for $f(x)$ and $f^{-1}(x)$ should mirror across the $y = x$ line.

**5. Label**:

- Write the equations next to their lines so you know which is which.

**Example 2: **

Finding and Sketching the Inverse of Quadratic Function $g(x) = x^2$ (with $x \geq 0$)

**Solution:**

**1. Draw Axes**: Create a vertical y-axis and a horizontal x-axis.

**2. Plot **$g(x) = x^2$:

- Start at the origin (0,0).
- As $x$ increases, square $x$ to find $y$ (e.g., if $x = 1$, then $y = 1^2 = 1$; if $x = 2$, then $y = 2^2 = 4.$.
- Sketch a curve that starts at the origin and opens upwards to the right.

**3. Plot **$g^{-1}(x) = \sqrt{x}$:

- Use the same $y$ values as $x$ values for $g(x)$ (e.g., if $y = 1$, $x = \sqrt{1} = 1$; if $y = 4$, $x = \sqrt{4} = 2)$.
- Sketch a curve mirroring the $g(x)$ curve in the lower half of the graph.

**4. Check Reflection**:

- The curves of $g(x)$ and $g^{-1}(x)$ should reflect over the line $y = x$.

**5. Label**:

- Clearly write $g(x) = x^2$ and $g^{-1}(x) = \sqrt{x}$ near the curves.

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.