Question

What is the inverse function of `f(x) = 2log_4 x`?

Answer 1

`f^"-1"(x)=4^(x//2)`.

Explanation:

For a function `f(x)`, the inverse function `f^"-1"(x)` satisfies `f(color(white)stackrel()color(black)f^"-1"(x))=x`. What this means is that if the input to `f` is the inverse function `f^"-1"(x)`, then `f` will return `x`.

In math terms, we start with:

`f(color(white)stackrel()color(black)f^"-1"(x))=x`

And since `f(*) = 2log_4(*)`, we get

`2log_4(f^"-1"(x))=x" "color(skyblue)[(star)]`

Now, we just solve for `f^"-1"(x)`:

`=>log_4(f^"-1"(x))=x/2`

`=>f^"-1"(x)=4^(x//2)`

And there's our inverse function.

Bonus:

The shortcut way to find inverse functions is to swap the positions of `x` and `f(x)` in your function, renaming `f(x)` to `f^"-1"(x)`, and solve for the new function. For this problem, that means:

`f(x)=2log_4 x`

becomes

`x=2log_4(f^"-1"(x))`

which is exactly the same as `color(skyblue)((star))` above.