How can I tell if a function has an inverse if it involves inverse trig functions?

Question

So I am trying to use this definition here

$f(x)=f(y)\Longrightarrow x=y $ to show that $f(x)=x^3+\tan^{-1}(x+1)$ has an inverse [since inverse functions must be 1-1 for an inverse to exist] but I can't get $x=y$ no matter what I try.

Am I doing something completely wrong or is there another way?

Answer 1

Trying with algebraic methods seems quite difficult in this case. We can try to see whether the function is monotonic; since $$ f'(x)=3x^2+\frac{1}{1+(x+1)^2} $$ you should easily be able to conclude.

If $f$ is strictly monotonic, $x<y$ implies $f(x)<f(y)$, so in particular that $x\ne y$ implies $f(x)\ne f(y)$.