In order to calculate g^-1(-9), since it's an inverse function, we just need to find the value of x for y = -9.
Looking at the set of points of g, we can see that for y = -9 we have x = 6, so:
[tex]g^{-1}(-9)=6[/tex]
Then, to find h^-1(x), let's change h(x) by x and x by h^-1(x) in the function:
[tex]\begin{gathered} h(x)=\frac{-x+13}{5} \\ x=\frac{-h^{-1}(x)+13}{5} \\ -h^{-1}(x)+13=5x_{}_{} \\ -h^{-1}(x)=5x-13 \\ h^{-1}(x)=13-5x \end{gathered}[/tex]
Finally, calculating (h o h^-1)(4), that is, the composite function h of h^-1 of 4 we have:
[tex]\begin{gathered} (hoh^{-1})(4)=h(h^{-1}(4)) \\ =h(13-5\cdot4) \\ =h(13-20) \\ =h(-7) \\ =\frac{-(-7)+13}{5} \\ =\frac{7+13}{5} \\ =\frac{20}{5} \\ =4 \end{gathered}[/tex]
