We have two functions f ( x ) and g ( x ) unknown to us.
We are given a mapped value of function f ( x ) as follows:
[tex]f\text{ ( -2 ) = 13}[/tex]
We are given that f ( x ) and g ( x ) are inverses of each other!
Recall, that an inverse of a function is reflection of given function about the following line:
[tex]y\text{ = x}[/tex]
Where,
[tex]\begin{gathered} x\text{ = input value} \\ y\text{ = output of the function} \end{gathered}[/tex]
Whenever we take out inverse of a function every input value ( x ) and the corresponding output of the function are interchanged with one another. This is expressed by the reflection about line ( y = x ).
So for the given data we have:
[tex]x\text{ = -2 , y = 13 : f ( x )}[/tex]
Since, g ( x ) is an inverse of f ( x ). We will interchange each value of x with each value of y. So for the above case we will express:
[tex]x=13,y=-2\colon g(x)=f^{-1\text{ }}\text{ ( x )}[/tex]
Then the corresponding value for the function g ( x ) would be:
[tex]\begin{gathered} g(13)=f^{-1}\text{ ( 13 ) = -2} \\ \textcolor{#FF7968}{g}\text{\textcolor{#FF7968}{ ( 13 ) = -2}} \end{gathered}[/tex]
Answer: Option D
