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CIE A-Level Maths Study Notes

1.2.5 Graphical Interpretation of Inverses

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=xy = 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=xy = 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)=yf(x) = y , then f1(y)=xf^{-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=xy = x. If a point (a,b)(a, b) lies on the graph of the function, then the point (b,a)(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=xy = x: Reflect each point of the original function across the line y=xy = 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=xy = x

The symmetry of a function and its inverse about the line y=xy = 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=xy = x is maintained, but the specific points of reflection shift. For example, translating f(x)f(x) up by 2 units will also shift f1(x)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=xy = x. For instance, reflecting f(x)f(x) across the x-axis will reflect f1(x)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=xy = x remains consistent, serving as a guide to understanding these changes.

Examples

Example 1:

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

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

3. Plot f1(x)=x32f^{-1}(x) = \frac{x - 3}{2}:

  • Mark points for xx 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)f(x) and f1(x)f^{-1}(x) should mirror across the y=xy = x line.

5. Label:

  • Write the equations next to their lines so you know which is which.
graph of inverse function

Example 2:

Finding and Sketching the Inverse of Quadratic Function g(x)=x2g(x) = x^2 (with x0x \geq 0)

Solution:

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

2. Plot g(x)=x2g(x) = x^2:

  • Start at the origin (0,0).
  • As xx increases, square xx to find yy (e.g., if x=1x = 1, then y=12=1y = 1^2 = 1; if x=2x = 2, then y=22=4.y = 2^2 = 4..
  • Sketch a curve that starts at the origin and opens upwards to the right.

3. Plot g1(x)=xg^{-1}(x) = \sqrt{x}:

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

4. Check Reflection:

  • The curves of g(x)g(x) and g1(x)g^{-1}(x) should reflect over the line y=xy = x.

5. Label:

  • Clearly write g(x)=x2g(x) = x^2 and g1(x)=xg^{-1}(x) = \sqrt{x} near the curves.
graph of quadratic function and its inverse
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

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.

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