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Finding the Inverse of a Function Using a Graph
(KS4, Year 10)
The Lesson
A function and its inverse function can be plotted on a graph. If the function is plotted as y = f(x), we can reflect it in the line y = x to plot the inverse function y = f^{−1}(x).Every point on a function with Cartesian coordinates (x, y) becomes the point (y, x) on the inverse function: the coordinates are swapped around.
How to Find the Inverse of a Function Using a Graph
Finding the inverse of a function using a graph is easy.Question
Find the inverse function of the function plotted below.StepbyStep:
1
Plot the function on a graph.
How to Plot a Function
The function is a linear equation and appears as a straight line on a graph.
The constant term gives the yintercept
In our example, the yintercept is 1. The line crosses the yaxis at 1.

The coefficient of the x term gives the slope of the line.
In our example, there is no number written in front of the x. It has an implicit coefficient of 1.
The line has a slope of 1. The line will go up by 1 when it goes across by 1.
2
Plot the line y = x on the same graph.
The line y = x is a 45° line, halfway between the xaxis and the yaxis.
The line y = x is a 45° line, halfway between the xaxis and the yaxis.
3
Reflect the line y = f(x) in the line y = x.
Each point on the reflected line is the same perpendicular distance from the line y = x as the original line.
The reflected line is the graph of the inverse function.
The reflected line is the graph of the inverse function.
4
Find the equation of the inverse function.

The yintercept is −1.

The slope is 1.
y = slope x + yintercept
y = 1x + −1
Answer:
The inverse of the function f(x) = x + 1 is: f^{−1}(x) = x − 1.What Is an Inverse Function?
An inverse function is a function that reverses another function. If a function f relates an input x to an output f(x)......an inverse function f^{−1} relates the output f(x) back to the input x:
Imagine a function f relates an input 2 to an output 3...
f(2) = 3
...the inverse function f^{−1} relates 3 back to 2...
f^{−1}(3) = 2
An inverse function is denoted f^{−1}(x).
How To Reflect a Function in y = x
To find the inverse of a function using a graph, the function needs to be reflected in the line y = x. By reflection, think of the reflection you would see in a mirror or in water:Each point in the image (the reflection) is the same perpendicular distance from the mirror line as the corresponding point in the object. If a function is reflecting the the line y = x, each point on the reflected line is the same perpendicular distance from the mirror line as the original function:
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