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:

1. Plot the Original Function: Begin by plotting the original function on a set of axes.
2. 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.
3. 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.