Answer:
If we have two functions [tex]f \ and \ g[/tex] such that [tex]f(g(x))=x[/tex] for every [tex]x[/tex] in the domain of [tex]g[/tex], and [tex]g(f(x))=x[/tex] for every for every [tex]x[/tex]in the domain of [tex]f[/tex]. If we prove this, then [tex]g[/tex] is the invers function of [tex]f[/tex] and denoted by [tex]f^{-1}[/tex]
1. We need to prove whether [tex]f(g(x))=x[/tex]. So:
[tex]f(x)=\frac{4}{5}x+1 \\ \\ g(x)=\frac{5x-5}{4} \\ \\ So: \\ \\ f(g(x))=\frac{4}{5}\left(\frac{5x-5}{4})+1 \therefore f(g(x))=\frac{4}{5}\left(\frac{5x-5}{4})+1[/tex]
[tex]\therefore f(g(x))=x-1+1 \\ \\ \boxed{f(g(x))=x}[/tex]
2. We need to prove whether [tex]g(f(x))=x[/tex]. So:
[tex]g(f(x))=\frac{5(\frac{4x}{5}+1)-5}{4} \\ \\ \\ g(f(x))=\frac{4x+5-5}{4} \\ \\ \\ g(f(x))=\frac{4x}{4} \\ \\ \\ g(f(x))=x[/tex]
Since [tex]f(g(x))=g(f(x))=x[/tex], then:
[tex]f(x) \ and \ g(x)[/tex] are inverses to each other.
