Determine whether each statement is true or false. If the stat...

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

If $f of x equals 3 x$ and $g (x) = x + 5$ , find ${(f \circ g)}^{- 1} (x)$ and $(g^{- 1} \circ f^{- 1}) (x)$ .

Accepted Answer

Welcome back. I'm so glad you're here. This is a fun one. We get to find a couple of things. We get to find F of G inverse of X. And we get to find g inverse of f inverse of X. And we are given that F of X equals five X. And that G of X equals x minus seven. So to work on the first one to work on F of G inverse of X. We first want to work on F of G of X and then we'll find the inverse of that. So F of G of X. We are going to take the G of X and stick that inside of our F of X wherever there is an X. So we start writing our F of X and we'll stop whenever we hit an X. So our F of X is five X. We start writing the five and then we stop for the X. And then we're going to substitute R G of X in for that X. So five times X minus seven. And we can simplify that, distributing the five. We've got five x minus 35. That's our F of G of X. Now to find the inverse of that. Well, whenever we're working on the inverse, what we're going to do is we're going to rewrite the F of G, the left hand side of the F of G inverse of X, we're going to write that as a Y. So Y equals and then we bring down our right hand side. So Y equals five X minus 35. And then we are going to switch our X. And our Y. So instead of Y equals its X equals. Instead of five X. It's five, Y minus 35. And now we're going to solve for why get that Y by itself the way it just wants to be. So we've got X plus 35 equals five. Y. After adding 35 to both sides. And then we will divide both sides by five. And leaving it just like this, Y equals X plus 35/5. And leaving it like that because that's the way the answers have it written. So the F of G inverse of X is X plus 35/5. That's one of our answers Now, for the next one. So we need to find the g inverse of f inverse of X. So the G of X. Rewriting it is x minus seven. We need to find the inverse of that. So again what we do is we take that left hand side and rewrite it as Y. Well here first. Alright, G inverse of X. What we're trying to find. So we rewrite the left hand side is Y. And bring down the right hand side equals x minus seven. And then we are going to swap our x and our Y. So instead of Y equals X equals and instead of x minus seven it's y minus seven. And we want to get the y by itself. So we'll add seven to both sides and we have X plus seven equals Y. Well, what's on the opposite side of the Y. This X plus seven, that is R G inverse of X. And now working on our after X that is five X. R f inverse of X will solve for that. We'll rewrite the left hand side as y bring down the right hand side. So Y equals five X. And then we'll swap our X. And our Y. So we now have instead of Y equals its X equals. And instead of five exits five, Y. We want to work to get why by itself the way it wants to be and we have Y equals X over five. And what's on the opposite side of the Y. That is our inverse. So f inverse of X is x over five. Okay, now to work on our g inverse of f inverse of X, we look and see that our f inverse of X is going to go inside of our g inverse of X. So we'll start writing our g inverse of X until we hit an X. And then we'll replace it with our f inverse of X. G inverse of X. Well, that starts off with an X. That's X plus seven. So instead of writing X there were going to write our f inverse of X, which is X over five. And then we continue writing r G inverse of X. So bringing down our plus seven and that is it. That is our g inverse of f inverse of X. We look at our answer choices and this matches with answer choice. A well done, we'll catch you on the next one.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If $f of x equals 3 x$ and $g (x) = x + 5$ , find ${(f \circ g)}^{- 1} (x)$ and $(g^{- 1} \circ f^{- 1}) (x)$ .

Key Concepts

Function Composition

Inverse Functions

Notation and Order of Operations in Compositions and Inverses

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