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Problem 37 Hard Difficulty

If you graph the function
$$ f(x) = \frac{1 - e^\frac{1}{x}}{1 + e^\frac{1}{x}} $$
you'll see that $ f $ appears to be an odd function. Prove it.

Answer

From the graph, it appears that $f$ is an odd function $(f \text { is undefined for } x=0$ )
To prove this, we must show that $f(-x)=-f(x)$
$\begin{aligned} f(-x) &=\frac{1-e^{1 /(-x)}}{1+e^{1 /(-x)}}=\frac{1-e^{(-1 / x)}}{1+e^{(-1 / x)}}=\frac{1-\frac{1}{e^{1 / x}}}{1+\frac{1}{e^{1 / x}}} \cdot \frac{e^{1 / x}}{e^{1 / x}}=\frac{e^{1 / x}-1}{e^{1 / x}+1} \\ &=-\frac{1-e^{1 / x}}{1+e^{1 / x}}=-f(x) \end{aligned}$
so $f$ is an odd function.

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Video Transcript

All right, So here's a function. We want to prove that it is odd. So remember that for an odd function, half of the opposite of X is equal to the opposite of F of X. So we're going to find f of the opposite of X and show that it is equal to the opposite of what we're given here. So f of the opposite of X is what we get when we substitute the opposite of X. Everywhere we have an X, so we're going to have one minus e to the one over the opposite of X power over one plus e to the one over the opposite of X power. Now our job is to simplify, simplify, simplify until we get it to look like where we want it or need it to look. Okay, so the next step is going to be to change these exponents. So when you see one over the opposite of X, that's the same as the opposite of one over X. Okay, you can have that negative sign in the front instead of in the denominator. So what we have here is one minus e to the negative one over X over one plus e to the negative one over X. Now, remember that a negative exponents is a reciprocal. So this is equivalent to one minus one over E to the one over X over one plus one over E to the one over X. Okay, well, now we've created quite a mess with fractions inside the fraction. So to simplify that, let's multiply the numerator and denominator by E to the one over X special form of one that's going to simplify things quite a bit. So we multiply each of the one over X by one, and that is each of the one over X. We multiply e to the one over X by one over each of the one over X, and that is one on the bottom. Same thing E to the one over X times one is e to the one over X plus and then each of the one over X times one over each of the one of her exes. One. Okay, Finally, when we look at that numerator, it is the opposite of one minus e to the one of Rex and the denominator. We could right by just reversing the order of the terms. Now let's take a look at what we have And let's take a look at what we started with are these opposites? They absolutely are opposites. So we have the opposite of F of X. So we just showed that f of the opposite of X is equal to the opposite of F of X and prove that f is indeed odd.