Question

please help !!!
Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = 8/x
and
g(x) = 8/x

Can you like really explain it though with words bc I want to make sure I understand well.

Answer 1

If you are looking at finding inverses, you should already know that only one-to-one "functions" have inverses.

A function is \(f: \mathbb R \to \mathbb R\). You do not have that here across the domain, singularity at zero

But if you want a simple approach, re-label the independent/dummy variable in the first "non-"function:

\(f(u) = \dfrac{8}{u}\)

Then call the second one \(u = g(x)\):

\( \implies u = g (x) = \dfrac{8}{x}\)

And so:

\(f(u) = \dfrac{8}{\frac{8}{x}} = x\)


Try explaining this via verbosity(?) .... defeats the object.

This is an example of involution, except they are not functions (!!)

Play with the idea, and make sure you know what a function is.

Answer 2

alright but that doesn't help me learn how to confirm that these are inverses. I don't even know what to do in this situation.

Answer 3

Doesn't help me either because I think you should have an idea what a function is before you try inverting it. But anyways.

If you take the first "function": \(f(x) = \dfrac{8}{x} \qquad = y\) <-- we're using that 'y' for a reason

Then re-write as: \(x = \dfrac{8}{y} \)

That's inverted, right, because we now plug in y values to get x values

So we say, for the inverted "function":

Then: \(f^{-1}(x) = \dfrac{8}{x} \)

Do the same for \(g(x)\) as they are the same function

Answer 4

\(x = \dfrac{8}{y}\) maps y to x

\(y = \dfrac{8}{x}\) maps x to y

Answer 5

im sorry I still don't think you understand the question but I guess thats okay i'll find someone else. Have a good day!