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.

Restrict the domain of the function so that the function is a onetoone function and has an inverse Find the inverse function State the domain and ranges of the class=

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[tex]\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}[/tex]