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`