ANSWER and EXPLANATION
We are given a function and its inverse function:
[tex]\begin{gathered} f(x)=\frac{1}{2}x \\ f^{-1}(x)=2x \end{gathered}[/tex]
To solve the problems, we have to substitute the values of x in the brackets into the appropriate function (or inverse function).
Therefore, we have that the value of the function for x = 2:
[tex]\begin{gathered} f(2)=\frac{1}{2}\cdot2 \\ f(2)=1 \end{gathered}[/tex]
For x = 1, we have that the value of the inverse function is:
[tex]\begin{gathered} f^{-1}(1)=2(1) \\ f^{-1}(1)=2 \end{gathered}[/tex]
For x = -2, we have that the value of the inverse function is:
[tex]\begin{gathered} f^{-1}(-2)=2\cdot-2 \\ f^{-1}(-2)=-4 \end{gathered}[/tex]
For x = -4, we have that the value of the function is:
[tex]\begin{gathered} f(-4)=\frac{1}{2}\cdot-4 \\ f(-4)=-2 \end{gathered}[/tex]
For the fifth option, substitute the value of the function at x = 2 into the inverse function.
That is:
[tex]\begin{gathered} f^{-1}(f(2))=f^{-1}(1)=2\cdot1 \\ f^{-1}(f(2))=2 \end{gathered}[/tex]
For the sixth option, substitute the value of the inverse function at x = -2 into the function.
That is:
[tex]\begin{gathered} f(f^{-1}(-2))=f(-4)=\frac{1}{2}\cdot-4 \\ f(f^{-1}(-2))=-2 \end{gathered}[/tex]
To find the general form of the function:
[tex]f^{-1}(f(x))=f(f^{-1}(x))[/tex]
either substitute the function for x in the inverse function or substitute the inverse function for x in the function.
Therefore:
[tex]\begin{gathered} f^{-1}(f(x))=2(\frac{1}{2}x)) \\ f^{-1}(f(x))=x \end{gathered}[/tex]
That is the answer.
