$$ \begin{array}{l} ext { The function } f(x)=(x+5)^{3} ext { is one-to-one. } \ ext { a. Find an equation for } f^{-1}(x) ext {, the inverse function. } \ ext { b. Verify that your equation is correct by showing that } f\left(f^{-1}(x) ight)=x ext { and } f^{-1}(f(x))=x ext {. } \ ext { a. Select the correct choice below and fill in the answer box(es) to complete your choice. } \ ext { (Simplify your answer. Use integers or fractions for any numbers in the expression.) } \ ext { B. } f^{-1}(x)=\square ext {, for all } x \ ext { C. } f^{-1}(x)=\square ext {, for } x \geq \square \ ext { D. for } x \leq \square\end{array} f^{-1}(x)=\square $$, for $$ x
eq \square $$,

Question

$$ \begin{array}{l}	ext { The function } f(x)=(x+5)^{3} 	ext { is one-to-one. } \ 	ext { a. Find an equation for } f^{-1}(x) 	ext {, the inverse function. } \ 	ext { b. Verify that your equation is correct by showing that } f\left(f^{-1}(x)ight)=x 	ext { and } f^{-1}(f(x))=x 	ext {. } \ 	ext { a. Select the correct choice below and fill in the answer box(es) to complete your choice. } \ 	ext { (Simplify your answer. Use integers or fractions for any numbers in the expression.) } \ 	ext { B. } f^{-1}(x)=\square 	ext {, for all } x \ 	ext { C. } f^{-1}(x)=\square 	ext {, for } x \geq \square \ 	ext { D. for } x \leq \square\end{array} f^{-1}(x)=\square $$, for $$ x 
eq \square $$,

UpStudy AI · Accepted Answer

1. Write the function with $$ y $$ in place of $$ f(x) $$:
   $$
   y = (x+5)^3
   $$

2. Swap $$ x $$ and $$ y $$ to find the inverse:
   $$
   x = (y+5)^3
   $$

3. Solve for $$ y $$ by taking the cube root on both sides:
   $$
   \sqrt[3]{x} = y+5
   $$

4. Isolate $$ y $$ by subtracting 5:
   $$
   y = \sqrt[3]{x} - 5
   $$

5. Therefore, the inverse function is:
   $$
   f^{-1}(x) = \sqrt[3]{x} - 5, \quad \text{for all } x.
   $$

6. Verify the result:

- Compute $$ f(f^{-1}(x)) $$:
     $$
     f\bigl(f^{-1}(x)\bigr) = \Bigl(\bigl(\sqrt[3]{x} - 5\bigr) + 5\Bigr)^3 = \bigl(\sqrt[3]{x}\bigr)^3 = x.
     $$
   
   - Compute $$ f^{-1}(f(x)) $$:
     $$
     f^{-1}\bigl(f(x)\bigr) = \sqrt[3]{(x+5)^3} - 5 = (x+5) - 5 = x.
     $$

The completed answer is:

- B. $$ f^{-1}(x)=\sqrt[3]{x}-5 $$, for all $$ x $$.
The inverse function is $$ f^{-1}(x) = \sqrt[3]{x} - 5 $$ for all $$ x $$.

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