Answer:
[tex]g(f(x))=\frac{-7(\frac{x-8}{x+7})-8}{(\frac{x-8}{x+7})-1}\\\\g(f(x))=x[/tex]
[tex]f(g(x))=\frac{(\frac{-7x-8}{x-1})-8}{(\frac{-7x-8}{x-1})+7}\\\\\\f(g(x))=x[/tex]
Step-by-step explanation:
First we write the function f(x)
[tex]f(x) = \frac{x-8}{x+7}[/tex]
Now we write the function g(x)
[tex]g(x)=\frac{-7x-8}{x-1}[/tex]
First we find [tex]f (g (x))[/tex]
To find [tex]f(g(x))[/tex] you must enter the function g(x) into the function f(x) by making [tex]x = g (x)[/tex]
[tex]f(g(x))=\frac{(\frac{-7x-8}{x-1})-8}{(\frac{-7x-8}{x-1})+7}[/tex]
Now we simplify the expression
[tex]f(g(x))=\frac{(\frac{-7x-8}{x-1})-8}{(\frac{-7x-8}{x-1})+7}\\\\\\f(g(x))=\frac{(\frac{-7x-8-8(x-1)}{x-1})}{(\frac{-7x-8+7(x-1)}{x-1})}\\\\\\f(g(x))=\frac{-7x-8-8(x-1)}{-7x-8+7(x-1)}\\\\\\f(g(x))=\frac{-7x-8-8x+8}{-7x-8+7x-7}\\\\\\f(g(x))=\frac{-15x}{-15}\\\\\\f(g(x))=x[/tex]
Now we find [tex]g(f(x))[/tex] doing [tex]x = f (x)[/tex]
[tex]g(f(x))=\frac{-7(\frac{x-8}{x+7})-8}{(\frac{x-8}{x+7})-1}[/tex]
Now we simplify the expression
[tex]g(f(x))=\frac{-7(\frac{x-8}{x+7})-8}{(\frac{x-8}{x+7})-1}\\\\\\\g(f(x))=\frac{(\frac{-7x+56}{x+7})-8}{(\frac{x-8-(x+7)}{x+7})}\\\\\\g(f(x))=\frac{(\frac{-7x+56-8(x+7)}{x+7})}{(\frac{x-8-(x+7)}{x+7})}\\\\\\g(f(x))=\frac{-7x+56-8(x+7)}{x-8-(x+7)}\\\\g(f(x))=\frac{-7x+56-8x-56}{x-8-x-7}\\\\g(f(x))=\frac{-15x}{-15}\\\\g(f(x))=x[/tex]
