What property must the function $g$ have such that if $\alpha = \beta$ where $g(a)=\alpha$, $g(b)=\beta$, then $a=b$.

Question

I am trying to prove that two functions are equivalent and I am required to use the function $g(x)=\cos{\frac{x}{55}}$ to prove that $h(x)=55\left(\frac{\pi}{2}+\arctan{(\sinh(4x)}\right)$ is equivalent to $f(x)=110\arctan{(e^{4x})}$, by considering $g(h(x))$ and $g(f(x))$. I have completed the algebra and have shown that $g(h(x))=g(f(x))$, but the question also asks which property of $g$ I have used for the proof. Any help is appreciated.

Answer 1

This is called injectivity; if $g(a) = g(b)$ implies $a=b,$ then we say $g$ is injective. Note that cosine is not injective over its entire domain, so be careful with how your proof works.