Question

Find f^{-1}(x) for the function below. Do not put spaces between your characters. If no inverse exists then type "none".f(x)=x-13f^{-1}(x)=Answer

Answer 1

Remember the definition of the inverse function

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

Then, in our problem, we need to find f^-1 such that

$\begin{gathered} f^{-1}(f(x))=x \\ \Rightarrow f^{-1}(x-13)=x \\ \end{gathered}$

Then, a clear possibility is:

$\begin{gathered} f^{-1}(y)=y+13 \\ \Rightarrow f^{-1}(x-13)=x-13+13=x \\ \Rightarrow f^{-1}(y)=y+13 \end{gathered}$

Now, the second condition

$\begin{gathered} f^{}(f^{-1}(y))=f(y+13)=(y+13)-13=y \\ \Rightarrow f^{}(f^{-1}(y))=y \end{gathered}$

Thus, the inverse function is x+13

$f^{-1}(x)=x+13$