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Problem 77 Hard Difficulty

(a) If we shift a curve to the left, what happens to its reflection about the line $ y = x $? In view of this geometric principle, find an expression for the inverse of $ g(x) = f (x + c) $, where $ f $ is a one-to-one function.
(b) Find an expression for the inverse of $ h(x) = f (cx) $ , where $ c \neq 0 $.


a) $g^{-1}(x)=f^{-1}(x)-c$
b) $h^{-1}(x)=\frac{1}{c} f^{-1}(x)$

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Video Transcript

Okay, let's explore this idea. So let's say we have some original curve. We're just going to keep it really simple. Let's say it was this function here. This is F of X, and maybe this point here is at a height of one. So it's inverse. The reflection across the line Y equals X would look like so, and it would have an X intercept at one, and I'll call it F in verse. Now what happens if I take my original curve and I shifted to the left? So let's say that my new curve looks like this. And at the point that used to be at a height of one, it was at 01 is now at, let's say negative 21 Okay, well, what will the inverse look like? I'm gonna call this one g. By the way, the inverse is going to go through the 0.1 negative too, and look like this. So notice that when I shifted the original one left, I ended up shifting the inverse down If ago left to with the original, I go down to with the inverse. So let's call this in. Verse G inverse of X So if the equation of G of x waas G of X equals half of X plus C and the plus C represents shifting left, then the equation of G inverse is going to be F in verse of X minus. See, the minus C on the outside indicates the shifting down. Now let's take a look at another case where we're going to have a horizontal stretch or shrink because we're multiplying by a number inside the function. So let's say our original curve was something like, Oh, this and its inverse. Even though the inverse isn't to function just for the sake of it of the exploration here, let's say the inverse in would have to look like this. Okay, so my original f and my f embers ignoring the fact that it's not a function just to explore the concept. Now suppose a stretch it horizontally and my new function looks like this. So that's my G. What will the new inverse look like? Well, I stretched my G horizontally, and that's going to result in a vertical stretch of the inverse. So there's G enders. So suppose my equation Fergie was G of X equals F of X f of C times X. The C represents the factor of horizontal stretch. It could be shrink to this is just it would apply the same way. So then my G in verse is going to have a vertical stretch or shrink depending on the situation. If I stretch horizontally, I'm going to stretch vertically. If I shrink horizontally and going to shrink vertically and so that would correspond toe one oversee times f of X f inverse of X. When you have a horizontal stretch, you end up with a C value greater than one inside the function. And when you have a vertical or yeah, horizontal stripes, you would actually have a C value less than one. And when you have a vertical stretch, you have a see value, the value right here greater than one. So one oversee would be greater than one