Question

A. Find an equation for F^-1 (x) the inverse function. B. Verify that your equation is correct by showing that (F ^-1 (x)) =x and f^-1 f(x))=x

Answer 1

A)

Replace y=f(x), isolate x and then replace x=f^-1(y) to find the inverse function:

$\begin{gathered} f(x)=x+7 \\ \Rightarrow y=x+7 \\ \Rightarrow y-7=x \\ \Rightarrow x=y-7 \\ \Rightarrow f^{-1}(y)=y-7 \\ \\ \therefore f^{-1}(x)=x-7 \end{gathered}$

B)

Evaluate f at f^-1(x) and f^-1 at f(x):

$\begin{gathered} f(f^{-1}(x))=f^{-1}(x)+7=x-7+7=x \\ f^{-1}(f(x))=f(x)-7=x+7-7=x \end{gathered}$

Therefore, the answer is option A)

$f^{-1}(x)=x-7\text{ for all x}$