Video Transcript
What is the inverse of the function 𝑦 equals seven 𝑥 minus five?
First, let’s consider what an inverse is. Formally, we could say if 𝑦 equals 𝑓 of 𝑥, then 𝑥 equals the inverse function evaluated at 𝑦. What’s happening here is that we’re switching the input and the output. For the function 𝑓 of 𝑥, the input values are 𝑥 and the output values are 𝑦. For the inverse, the input values will be 𝑦 and the output values will be 𝑥.
There’s an algebraic method we sometimes use for finding this function. This process involves two steps. First, we exchange the variables. That means we switch the 𝑥 and 𝑦. And in step two, we rearrange to express 𝑦 in terms of 𝑥. We know that 𝑦 equals seven 𝑥 minus five. But for our inverse function, 𝑥 is going to equal seven 𝑦 minus five. This completes step one.
However, we still prefer to write the equation so that it is in the form 𝑦 equals some 𝑥-value. And that means we want to rearrange this equation so the 𝑦 is by itself. To do that, we add five to both sides of the equation, and we now have 𝑥 plus five equals seven 𝑦. So we divide both sides of the equation by seven. On the left, we have 𝑥 plus five over seven. And on the right, we have 𝑦 by itself, which completes our second step and makes our inverse function 𝑦 equals 𝑥 plus five over seven.
