Question Video: Recognizing Inverse Functions by Composition Mathematics

Video Transcript

Let 𝑓 of 𝑥 equal two 𝑥 minus one divided by three 𝑥 minus two. Determine which of the following functions 𝑔 is the inverse of 𝑓 by checking 𝑓 of 𝑔 of 𝑥 equals 𝑥.

So essentially, we will be taking 𝑔 of 𝑥 and plugging it into 𝑓 of 𝑥 and if we get 𝑥 as our final answer, that will be our inverse. So let’s first check a). So here we have our function 𝑓 of 𝑥. And now we are going to take 𝑔 of 𝑥 for option a) and plug it into our 𝑓 of 𝑥 functions. We’ll plug that in for 𝑥.

So here we can see that we’ve plugged in two 𝑥 minus one divided by three 𝑥 minus two in for 𝑥 on the numerator and denominator. Now we need to distribute. So here we can see that we’ve distributed.

Now I need to subtract these fractions on the numerator and subtract the fractions on the denominator. So one or minus one is the same as three 𝑥 minus two over three 𝑥 minus two. So we’ve replaced one with three 𝑥 minus two over three 𝑥 minus two and we can replace two with six 𝑥 minus four divided by three 𝑥 minus two. And now we’re subtracting these fractions. And when you subtract fractions, you keep the denominator and you combine the numerator. So we need to be careful of these minus signs.

Now we need to combine like terms. We get 𝑥 divided by three 𝑥 minus two on the numerator and one divided by three 𝑥 minus two in the denominator. Now when dividing fractions, we will take the bottom, the denominator, and we will flip and multiply. So we’ll multiply by the reciprocal of the denominator. So here we can see that we’ve kept our numerator and then we’re multiplying by the reciprocal of the denominator. The three 𝑥 minus two is cancelled and we’re left with 𝑥, which is what we wanted. So that means 𝑔 of 𝑥 equals two 𝑥 minus one divided by three 𝑥 minus two would be the inverse of 𝑥.