Question

How do you find `f^{-1}(c)` if `f(x)= 5x + 8 x^{11}`; c = -13?

Answer 1

`f(x)=8x^11+5x`

Let
`n = f^(-1)(-13)`.
`f(n)=-13`

We have to check where `f(x)` is equal to `-13`.
`-13=8x^11+5x`

After graphing, I found that `f(x)` is equal to `-13` at `x=-1`.

Answer 2

`f^(-1)(-13)=-1`

Explanation:

In general the given function is in the form `y=f(x)` and its inverse is `f^(-1)(y)=x`, provided that `f(x)` is always crescent or decrescent for each value in the domain. This is just the case, being `f'(x)=5+11x^(10)` always positive for any `x in R`.

checked this, the problem of finding `f^(-1)(-13)=x` can be seen in an equivalent way as the one of finding the value of `x` whose image is `-13`.

To find it, it is enough to solve the equation `8x^(11)+5x-13=0`.
This has one root in `x=-1` so that at this point in the domain corresponds the value `y=-13` in the image. Being `f(-1)=13` and verified its monotonicity, we can concude that `f^(-1)(-13)=-1`