Let $f(x) = (x+2)^2-5$. If the domain of $f$ is all real numbers, then $f$ does not have an inverse function, but if we restrict the domain of $f$ to an interval $[c,\infty)$, then $f$ may have an inverse function. What is the smallest value of $c$ we can use here, so that $f$ does have an inverse function?
+26367 Let\( f(x) = (x+2)^2-5\).
If the domain of \(f\) is all real numbers,
then \(f\) does not have an inverse function,
but if we restrict the domain of \(f\) to an interval \([c,\infty)\),
then \(f\) may have an inverse function.
What is the smallest value of \(c\) we can use here,
so that \(f\) does have an inverse function?
\(\begin{array}{|rcll|} \hline \mathbf{f(x)} &=& \mathbf{(x+2)^2-5} \\ \\ x &=& \Big(f^{-1}(x)+2\Big)^2-5 \quad | \quad \text{inverse function} \\ x+5 &=& \Big(f^{-1}(x)+2\Big)^2 \\ \Big(f^{-1}(x)+2\Big)^2 &=& x+5 \\ f^{-1}(x)+2 &=& \pm\sqrt{x+5} \\ \mathbf{f^{-1}(x)} &=& \mathbf{-2 \pm\sqrt{x+5}} \\ \hline \\ \sqrt{x+5} &=& 0 \quad | \quad \text{the smallest value of }x \text{ is } c \\ x+5 &=& 0 \\ x &=& -5 \quad | \quad \text{the smallest value is } -5 \\ \\ \hline \mathbf{[c,\infty)} &=& \mathbf{[-5,\infty)} \\ \hline \end{array}\)
