The function $$ f(x)=x^{3}+10 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x) ight)=x $$ and $$ f^{-1}(f(x))=x $$. a. Select the correct choice below and fill in the answer box(es) to complete your choice (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ B. $$ f^{-1}(x)=\square $$, for $$ x \geq \square $$ C. $$ f^{-1}(x)=\square $$, for $$ x
eq \square $$ D. $$ f^{-1}(x)=\square $$, for all $$ x $$

Question

The function $$ f(x)=x^{3}+10 $$ is one-to-one. a. Find an equation for $$ f^{-1} $$, the inverse function. b. Verify that your equation is correct by showing that $$ f\left(f^{-1}(x)ight)=x $$ and $$ f^{-1}(f(x))=x $$. a. Select the correct choice below and fill in the answer box(es) to complete your choice (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. $$ f^{-1}(x)=\square $$, for $$ x \leq \square $$ B. $$ f^{-1}(x)=\square $$, for $$ x \geq \square $$ C. $$ f^{-1}(x)=\square $$, for $$ x 
eq \square $$ D. $$ f^{-1}(x)=\square $$, for all $$ x $$

UpStudy AI · Accepted Answer

**Step 1. Find the Inverse Function**

Given the function  
$$
f(x) = x^3 + 10,
$$  
we set  
$$
y = x^3 + 10.
$$  
To find the inverse, solve for $$x$$ in terms of $$y$$. First, isolate the $$x^3$$ term:  
$$
x^3 = y - 10.
$$  
Then take the cube root of both sides:  
$$
x = \sqrt[3]{y - 10}.
$$  
Thus, the inverse function is:  
$$
f^{-1}(y) = \sqrt[3]{y - 10}.
$$

Rewriting with the standard input variable $$x$$ we have:  
$$
f^{-1}(x) = \sqrt[3]{x - 10}.
$$

Since $$f(x)$$ is one-to-one (being a strictly increasing cubic function) and its domain and range are both all real numbers, the inverse is defined for all $$x$$. Therefore, the correct choice is:

D. $$\, f^{-1}(x)=\sqrt[3]{x-10}$$, for all $$x$$.

---

**Step 2. Verify the Inverse**

1. **Verify $$ f\left(f^{-1}(x)\right)=x $$:**

Start with:  
   $$
   f\left(f^{-1}(x)\right) = f\left(\sqrt[3]{x-10}\right).
   $$
   Substitute into $$f$$:  
   $$
   f\left(\sqrt[3]{x-10}\right) = \left(\sqrt[3]{x-10}\right)^3 + 10.
   $$
   Since $$\left(\sqrt[3]{x-10}\right)^3 = x-10$$, it follows that:  
   $$
   \left(\sqrt[3]{x-10}\right)^3 + 10 = (x-10) + 10 = x.
   $$

2. **Verify $$ f^{-1}(f(x))=x $$:**

Start with:  
   $$
   f^{-1}\left(f(x)\right) = f^{-1}\left(x^3+10\right).
   $$
   Substitute into $$f^{-1}$$:  
   $$
   f^{-1}\left(x^3+10\right) = \sqrt[3]{(x^3+10)-10}.
   $$
   Simplify the expression inside the cube root:  
   $$
   \sqrt[3]{x^3+10-10} = \sqrt[3]{x^3} = x.
   $$

Since both compositions yield $$x$$, the inverse function is verified.

---

**Final Answer**

D. $$\, f^{-1}(x)=\sqrt[3]{x-10}$$, for all $$x$$.
The inverse function is $$ f^{-1}(x) = \sqrt[3]{x - 10} $$ for all $$ x $$.

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