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

Nov 11, 2016

$f \left(x\right) = 8 {x}^{11} + 5 x$

Let
$n = {f}^{- 1} \left(- 13\right)$.
$f \left(n\right) = - 13$

We have to check where $f \left(x\right)$ is equal to $- 13$.
$- 13 = 8 {x}^{11} + 5 x$

After graphing, I found that $f \left(x\right)$ is equal to $- 13$ at $x = - 1$.

Nov 11, 2016

${f}^{- 1} \left(- 13\right) = - 1$

#### Explanation:

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

checked this, the problem of finding ${f}^{- 1} \left(- 13\right) = x$ 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 $8 {x}^{11} + 5 x - 13 = 0$.
This has one root in $x = - 1$ so that at this point in the domain corresponds the value $y = - 13$ in the image. Being $f \left(- 1\right) = 13$ and verified its monotonicity, we can concude that ${f}^{- 1} \left(- 13\right) = - 1$