How do I find the inverse function of a polynomial with $x^5$?

Question

I've been stumped on this problem for hours and cannot figure out how to do it from tons of tutorials.

Please note: This is an intro to calculus, so we haven't learned derivatives or anything too complex.

Here's the question:

Let $f(x) = x^5 + x + 7$. Find the value of the inverse function at a point. $f^{-1}(1035) = $___?

I tried setting $f(x)$ as $y$.. and solving for $x$. Clearly that doesn't help lol. I've tried many different approaches and cannot figure out the answer. I used wolframalpha, my textbook, notes, examples, and tons of Google searches and nothing makes sense. Can someone please help? Thanks!!

Answer 1

$$1035-7=1028=1024+4=4^5+4$$ Therefore $f^{-1}(1035)=4$.

Answer 2

HINT(s)

1. $f$ is an increasing function.
2. Since $f$ is increasing, you will be able to modify your guesses to close in on the answer quickly.

Answer 3

In general, polynomials won't have an inverse. This one happens to have one, but it's not fun to express, as far as I know.

Since you only need to find the inverse at a particular number, not any $y$, just plug it in and rearrange until something looks nice: $x^5 + x + 7 = 1035$ means $x(x^4 + 1) = 1028$. The factors of $1028$ are a good place to start.