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Prove f,-1" f, ntl -f = (-1)" Vnz2...

Question

Prove f,-1" f, ntl -f = (-1)" Vnz2

Prove f,-1" f, ntl -f = (-1)" Vnz2



Answers

$$ \text { If } f(x)=x^{2}-\frac{1}{x^{2}}, \text { prove that } f(x)=-f\left(\frac{1}{x}\right) \text { . } $$

Okay. Hello? So here we have, the F of X is equal to one over X and G of X is also equal to one over X. So if we compose these two functions, the same function in this case, what do we get? Well, if we do have composed G that's going to be F of G of X. So we input G and F. So we have one over or where we have an X. We input Gs we have 1/1 over X or 1/1 over X. It's just one times X over one which is equal to X. And if we did it the other way, well obviously it's gonna be the same, right? Because this is the same function. So if we did G composed F, that's going to be G of F of X. Where now we're inputting F and G. But since we have the same exact function, we're going to get the exact same thing, we get 1/1 over X. Which again is equal to just, well, again, one times, you know, X over one, which again is equal to X. So, um so in this case, right one over X is actually the inverse of itself. Right. Both cases we get the F composed G is equal to decompose F. And in both cases we get back to X.

All right, so we're composing one over X with itself. This is an interesting case that they are asking us. Hey, when this happens, you should get acts out. So verify it. We have f of one over X. So that's like 1/1 over x. And, uh, what we see here, this one we can rewrite as 1/1 if we wanted to and see that it's 1/1 times X over one. And that's just a fancy way of saying one Times X, which is X. And since GF is actually gonna be the same thing, we see that the reciprocal function is the one and only function besides the identity of function that you can compose with itself and get acts out. So it's its own inverse.

We know the M E is the medium. Therefore, m E is gonna be 1/2. Do you see a friend is also the median. Therefore we know FN is also 1/2 D c. Therefore, we noto um and is a B plus D c over too. Therefore, this indicates that we have e f equals seven minus dese malice M e minus half an. Therefore, we have a B minus D c over too.

If F of X was equal to a to the eggs, then F of negative X would equal a to the negative. X A negative power is the reciprocal so that be equal to one over a to the X, which is equal to one over f of X.


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