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If f(x)=69-15x then [tex]f(f^{-1}(x))=x[/tex]
Given :
[tex]f(x)=69-15x\\[/tex]
Lets find out [tex]f^{-1}(x)[/tex]. The steps to find out f inverse are
Replace f(x) by y
[tex]y=69-15x\\Interchange \; the \; variables\\x=69-15y\\solve \; for \; y\\x-69=-15y\\Divide \; by \; -15\\\frac{-x+69}{15} =y\\Replace \;y \;with \; f^{-1}\\f^{-1}=\frac{-x+69}{15}[/tex]
Now we find [tex]f(f^{-1}(x))[/tex]
Replace inverse function in the place of x in f(x)
[tex]f(f^{-1}(x))=f(\frac{-x+69}{15} )=69-15(\frac{-x+69}{15})=69-(-x+69)=69+x-69=x\\f(f^{-1}(x))=x[/tex]
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