A function is an equation that relates one variable (known as the independent variable) to another variable, known as the dependent variable. It’s an essential part of any maths lesson, but there are lots of different ways to learn about them.
This article explores a very simple way to show the function f(x) and its inverse by reflecting the graph of f across the line y = x and swapping the roles of x and y. By doing this, we can see that the inverse of a function is not simply a list of points, but it is actually a graph in its own right.
To prove this, we’ll look at a couple of points on each of these graphs. The point (0, 0) on the left is the image of the y-intercept of f(x), and so lies on the line y=x. The same is true for the point (1, 0) on the right.
If we substitute in these values, we can see that the inverse of the first function is indeed a graph and passes the horizontal line test for functions; however, the inverse of the second function does not pass this test, as it contains two points that have the same y-value, meaning they are not equal to one another. This is why it’s important to restrict the domain of a function when finding its inverse, as this will ensure that the inverse will also be a graph.