a) To find the inverse function, first, we set the following equation:
[tex]y=7x+2.[/tex]
Now, we solve the above equation for x:
[tex]\begin{gathered} y-2=7x, \\ \frac{y-2}{7}=x\text{.} \end{gathered}[/tex]
Finally, we exchange x and y
[tex]y=\frac{x-2}{7},[/tex]
and set y=f(x). Therefore,
[tex]f^{-1}(x)=\frac{x-2}{7}.[/tex]
b) To verify that the above function is the inverse, we compute:
[tex]f(f^{-1}(x))andf^{-1}(f(x)).[/tex]
For f(f^-1(x)), we get:
[tex]f(f^{-1}(x))=f(\frac{x-2}{7})=7\frac{x-2}{7}+2=x-2+2=x.[/tex]
For f^-1(f(x)) we get:
[tex]f^{-1}(f(x))=f^{-1}(7x+2)=x.[/tex]
Answer:
Part A:
[tex]f^{-1}(x)=\frac{x-2}{7}.[/tex]
Part B:
[tex]f(f^{-1}(x))=f(\frac{x-2}{7})=x.[/tex][tex]f^{-1}(f(x))=f^{-1}(7x+2)=x.[/tex]
