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

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.

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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.