# Inverse of a function involving trigonometry?

$g(x)=4-3\sin x$ is defined for the domain $x \in [\pi/2,A].$ The question is to find the largest value of $A$ for which $g$ has an inverse. I was thinking to find the inverse of $g(x)$, and find the range of $g(x)$ since the range of the inverse=domain of the original function, but I didn't get the right answer.

• It's better to do it generally: When does a function not have an inverse function? May 2 '17 at 13:46
• @Arthur when it's one on one? May 2 '17 at 13:52
• @JohnFire It needs to be onto too. May 2 '17 at 13:54
• @JohnFire Here is mathjax tutorial to format equations. May 2 '17 at 13:58
• Should be $(3/2) pi$ then @Arthur May 2 '17 at 14:02

The inverse is intended to be a function. This means it cannot take on multiple values as the "vertical line test." What this means for $g(x)$ is that it cannot take on multiple values as a "horizontal line test" otherwise its inverse $g^{-1}(x)$ would fail the vertical line test. Working through this this will give you the interval.
• of fives Does that mean $x^2$ is not a function then? May 2 '17 at 14:05
• @JohnFire No, it means that $x^2$ has no inverse function on the real line. It does have an inverse function if considered on $[0, \infty)$, however. May 2 '17 at 14:08