Question

Restrict the domain of the function so that the function is a one-to-one function and has an inverse. Find the inverse function, State the domain and ranges of the function and its inverse. Explain your results.

Answer 1

`\begin{gathered} \text{restrict the function }f(x)=(x+6)^2 \\ \text{The vertex is located at }(-6,0),\text{ therefore, this is the point where it is symmetrical} \\ \text{We can restrict the domain to:} \\ \text{Domain: }\lbrack-6,\infty) \\ \text{Range: }\lbrack0,\infty)| \\ \text{To find the inverse, replace y with x and vice versa} \\ f(x)=(x+6)^2 \\ y=(x+6)^2,\text{ then do the replacement} \\ x=(y+6)^2 \\ \sqrt[]{x}=\sqrt[]{(y+6)^2},\text{ then get the square root} \\ \sqrt[]{x}=y+6 \\ \sqrt[]{x}-6=y \\ y=\sqrt[]{x}-6 \\ \text{therefore, the inverse is} \\ f^(-1)(x)=\sqrt[]{x}-6 \\ \text{the Domain and range of the inverse is} \\ \text{Domain: }\lbrack0,\infty) \\ \text{Range: }\lbrack-6,\infty) \\ \text{As observed, the domain of the original function is the range of the inverse function} \\ \text{whereas, the range of the original function is the domain of the inverse function} \end{gathered}`