Question
For f: R→R and g: R→Ras the functions defined by the rulesf(x) =x2-2x for every x∈R and g(y) =x+1for all y∈R, show that f and g are inverse functions of one anotherusing composition
For f: R→R and g: R→R as the functions defined by the rules f(x) =x2-2x for every x∈R and g(y) =x+1 for all y∈R, show that f and g are inverse functions of one another using composition
Answers
Verify that $f$ and $g$ are inverse functions using the composition property. $$f(x)=x^{2}+1, x \geq 0, g(x)=\sqrt{x-1}$$
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So if we want to show that in versus or unique to actually give us a hint of assuming that both G and H R in versus of F and then they wanted us to look at a particular composition being G composed with F composed with h of X. So let's go ahead and go with this here. So first notice. Since F and H are going to be in verses, this is just going to be equal to G of X when we can just say since yes and h are in verses and then we can also right that this is going to be just equal to h of x sense. We're assuming F and G are in verses. But now this implies that h of X is equal to G of X, which would then imply that, um, our inverse so in verse is unique. Since no matter how we do this, we always end up getting that dysfunction. H of X has to be equal to G of X, so they're just the same function. So now that you've finished your proof, you could put your little proof box and smiling face because you're glad you're done with it.
Alright, here we go. So, for this one we've got two functions that are identical. F of X is one of our X in G of X is one of our X. And it wants you to show that F of X and G of X are in verses of one another. Okay. And so to show that we have to show that F of G of X is X, and the G of F of X is X. But since F of G of X and sorry, F of X and G of X are the same function. We really really need to show one. Right, Okay, so let's do this bottom one just because it's right there. So what is G of F of X? Well, that's G of F of X is one over X. So that's one over one over X. So if I multiply the bottom by X and the top by X on the bottom those cancel out. And on the top we're left with X. There you go. All right. And then to sketch him, well, the graph of F of X and G of X obviously are the same. And if you sketch those graphs, they just kind of look like this. It's like, you know what, let me do it a little bit better. You know that the inverse of one is one. So it's going to look something like that. And similarly the inverse of negative one is negative one. So it's going to look something like that. And again, it's it's just showing you remember that. Yeah, for a given function, the inverse of that function is just the reflection across the line shoot. I wanted that to be different color. Sorry, let me do that in yellow. So this yellow line here is the line Y equals X. And sort of find the inverse of a function. You just reflect the function across Y equals ax. But you can see here if you do that, you just get itself because this function is symmetrical about that line. All right, there we go.