Answer:
[tex]f^{-1}(g^{-1}(x))=\dfrac{1}{5}x+2.[/tex]
Step-by-step explanation:
1. If [tex]g(x)=x-4,[/tex] then
[tex]y=x-4,\\ \\x=y+4.[/tex]
Change y into x and x into y:
[tex]y=x+4,\\ \\g^{-1}(x)=x+4.[/tex]
2. If [tex]f(x)=5x-6,[/tex] then
[tex]y=5x-6,\\ \\5x=y+6,\\ \\x=\dfrac{y+6}{5}.[/tex]
Change y into x and x into y:
[tex]y=\dfrac{x+6}{5},\\ \\f^{-1}(x)=\dfrac{x+6}{5}.[/tex]
3. Find [tex]f^{-1}(g^{-1}(x)):[/tex]
[tex]f^{-1}(g^{-1}(x))=f^{-1}(x+4)=\dfrac{(x+4)+6}{5}=\dfrac{x+10}{5}=\dfrac{1}{5}x+2.[/tex]
