### 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.