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CIE A-Level Maths Study Notes

1.2.4 Inverse Functions

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


An inverse function, denoted as f1(x)f^{-1}(x), reverses the action of a function. For example, if f(3)=5f(3) = 5, then f1(5)=3f^{-1}(5) = 3. The graphs of y=f(x)y = f(x) and y=f1(x)y = f^{-1}(x) are symmetrical about the line y=xy = x.


A fundamental property of inverse functions is:

f(f1(x))=f1(f(x))=xf(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


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

2. Rearrange as ( y ): Express f(x)f(x) as yy.

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

4. Swap Variables: Replace each xx with yy.

5. Represent Inverse Function: Substitute yy with f1(x)f^{-1}(x).

Example 1

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

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

2. Rearrange for xx: x=y43x = \frac{y - 4}{3}.

3. Swap xx and yy: y=x43.y = \frac{x - 4}{3}.

4. Inverse notation: f1(x)=x43f^{-1}(x) = \frac{x - 4}{3}.

Example 2

For f(x)=2x5f(x) = 2x - 5:

1. Express as yy: y=2x5.y = 2x - 5.

2. Solve for xx: x=y+52x = \frac{y + 5}{2}.

3. Interchange xx and yy: y=x+52y = \frac{x + 5}{2}.

4. Represent inverse: f1(x)=x+52f^{-1}(x) = \frac{x + 5}{2}.

Example 3

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

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

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

3. Replace xx with yy: y=1xy = \frac{1}{x}.

4. Inverse form: f1(x)=1xf^{-1}(x) = \frac{1}{x}.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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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