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

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!!

• Guess and check! There is no nice formula for solving a fifth degree equation (that's a theorem). Your function is increasing, so there is a unique solution to $f(x)=1035$. If you fool around, you will find it very fast. – André Nicolas Sep 10 '13 at 4:20
• Oh no, André edited his comment as I wrote my answer, and now they're the same. All credit to André, the comment FGITW. – davidlowryduda Sep 10 '13 at 4:24

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

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.
• So it's literally a guess and check type ordeal? – Cozen Sep 10 '13 at 4:24
• @Justin: Why not? – davidlowryduda Sep 10 '13 at 4:25
• Since $f$ is rapidly increasing, this will converge quickly. Take $x_0=1035^{\frac 15}$, ignoring the other terms. Then $x_i=(x_{i-1}^5-x_{i-1}-7)^{\frac 15}$ – Ross Millikan Sep 10 '13 at 4:32

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.

• Ah, that's a nice one too. – davidlowryduda Sep 10 '13 at 4:26