Consider the function f : R-->R defined by f(x) =x^7+x+1. Show that f has an inverse

Question

We just started doing inverse functions so I'm not very familiar with this concept...

Answer 1

$f'(x)=7x^6+1>0$ implies that $f$ is strictly increasing thus injective. $lim_{x\rightarrow-\infty}f(x)=-\infty$, $lim_{x\rightarrow+\infty}f(x)=+\infty$ implies that $f$ is surjective since for every $n>0$, there exists $a<-n<n<b$ such that $f(x)=a$, $f(y)=b$ and $[a,b]$ is in the image of $f$ since $f$ is continue and the image of $R$ is connected.

Answer 2

$f(x)$ is a continuos function and the derivative $f'(x)=7x^6+1>0$ so the inverse function exists because the function is strictly increasing.