Question Video: Finding the Inverse of a Linear Function Algebraically Mathematics • Second Year of Secondary School

Video Transcript

Find 𝑓 inverse of 𝑥 for 𝑓 of 𝑥 equals a half 𝑥 plus three.

We’ve been given a function 𝑓 of 𝑥 and asked to find an expression for its inverse. We begin by letting 𝑦 equal 𝑓 of 𝑥. We then want to rearrange this equation to make 𝑥 the subject. The first step is to subtract three from each side, giving 𝑦 minus three equals a half 𝑥. We can then multiply both sides of the equation by two, giving a two multiplied by 𝑦 minus three is equal to 𝑥. And distributing the parentheses on the left-hand side, we have two 𝑦 minus six is equal to 𝑥. Now, this is 𝑥 as a function of 𝑦, but we want to write the inverse function as a function of 𝑥. We therefore swap the positions of 𝑥 and 𝑦 around. So where we had 𝑦, we swap this for 𝑥, and where we had 𝑥, we change this for 𝑦. So we now have 𝑦 is equal to two 𝑥 minus six.

The final step is to define this new function of 𝑥 to be 𝑓 inverse of 𝑥. So we have that 𝑓 inverse of 𝑥 is equal to two 𝑥 minus six. This is the function that maps all of the old output values back to their original inputs. We can see this if we take an input value, for example, four. When we apply the function 𝑓 of 𝑥 to the value four, we get the output value five. And when we apply this inverse function that we’ve just found to the output value five, we have that 𝑓 inverse of five is equal to two multiplied by five minus six, that’s 10 minus six, which gives the original input value of four.

It’s also worth pointing out that the two steps of rearranging to make 𝑥 the subject and then swapping 𝑥 and 𝑦 around can be done in either order. If you prefer, you can swap 𝑥 and 𝑦 at the beginning and then rearrange to make 𝑦 the subject. We’ll get the same answer, which is that 𝑓 inverse of 𝑥 is equal to two 𝑥 minus six.