Answer:
[tex]g^{-1}(x)=-\frac{3}{2}x-\frac{15}{2}[/tex]
Step-by-step explanation:
Given function,
[tex]g(x) = -\frac{2}{3}x-5[/tex]
Step 1 : Replace g(x) by y:
[tex]y = -\frac{2}{3}x-5[/tex]
Step 2 : Swap x and y:
[tex]x = -\frac{2}{3}y-5[/tex]
Step 3 : Solve the equation for y ( isolate y in the left side ):
[tex]x +\frac{2}{3}y=-5[/tex]
[tex]\frac{2}{3}y=-5-x[/tex]
[tex]y=\frac{3}{2}(-5-x)[/tex]
[tex]y=-\frac{15}{2}-\frac{3}{2}x[/tex]
[tex]y=-\frac{3}{2}x-\frac{15}{2}[/tex]
Step 4: Replace y by [tex]g^{-1}(x)[/tex]:
[tex]g^{-1}(x)=-\frac{3}{2}x-\frac{15}{2}[/tex]
Hence, the inverse of the function g(x) is [tex]g^{-1}(x)=-\frac{3}{2}x-\frac{15}{2}[/tex].
