Given the function \(g(x) = -\frac{{1}}{{2}}x^2+4x\) on the domain \(x\geq 4\), evaluate \(g^{-1}(-4.5)\).
Solution: When evaluating an inverse function, there are 2 approaches. Either solve for the inverse function algebraically and then substitute the input value or set the original function equal to value you wish to evaluate. I will use the latter.
\[\begin{{array}}{{rcl}} g(x)&=&=-\frac{{1}}{{2}}x^2+4x\\ -4.5&=&-\frac{{1}}{{2}}x^2+4x\\ 9 &=&x^2 -8x\\ 0&=&x^2-8x-9\\ 0&=&(x-9)(x+1)\\ x=9&\text{{or}}&x=-1 \end{{array}} \] Since the domain is restricted to \(x\geq 4\), the answer is \(x=9\).