Question

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Answer 1

Answer:

If we have two functions $f \ and \ g$ such that $f(g(x))=x$ for every $x$ in the domain of $g$, and $g(f(x))=x$ for every for every $x$in  the domain of  $f$. If we prove this, then $g$ is the invers function of $f$ and denoted by $f^{-1}$

1. We need to prove whether $f(g(x))=x$. So:

$f(x)=\frac{4}{5}x+1 \\ \\ g(x)=\frac{5x-5}{4} \\ \\ So: \\ \\ f(g(x))=\frac{4}{5}\left(\frac{5x-5}{4})+1 \therefore f(g(x))=\frac{4}{5}\left(\frac{5x-5}{4})+1$

$\therefore f(g(x))=x-1+1 \\ \\ \boxed{f(g(x))=x}$

2. We need to prove whether $g(f(x))=x$. So:

$g(f(x))=\frac{5(\frac{4x}{5}+1)-5}{4} \\ \\ \\ g(f(x))=\frac{4x+5-5}{4} \\ \\ \\ g(f(x))=\frac{4x}{4} \\ \\ \\ g(f(x))=x$

Since $f(g(x))=g(f(x))=x$, then:

$f(x) \ and \ g(x)$ are inverses to each other.