I would be in need of a little help with a problem im facing. I have to show that the function
$$ f(x) = x^7 + x^²$$
defined at $f: \Bbb{R} $ $\rightarrow$ $\Bbb{R}$ has a continuous inverse function $g: \Bbb{R} $ $\rightarrow$ $\Bbb{R}$
How would this go? Finding the inverse of the particular function showed out to be relatively tricky to find.
In order for a function $f$ to have an inverse $f^{-1}$ is being bijective, which means being 1-1 and onto. We need to check whether the function you gave is bijective, so we can simply graph it:
As you can see, $f(x)=x^7+x^2$ is not 1-1, as it does not meet the definition: $$f(a)=f(b) \Rightarrow a = b$$ which means that $f$ does not have an inverse function.