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 choic (Simplify your answer. Use integers or fractions for any numbers in the expression.) B. $$ f^{-1}(x)=\square $$, for $$ x
eq \square $$ C. $$ f^{-1}(x)=\square $$, for all $$ x $$ D. for $$ x \leq \square $$ b. Verify that the equation is correct. $$ f(x)=\square $$, for $$ x \geq \square(x))=f(\square) $$

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 choic (Simplify your answer. Use integers or fractions for any numbers in the expression.) B. $$ f^{-1}(x)=\square $$, for $$ x 
eq \square $$ C. $$ f^{-1}(x)=\square $$, for all $$ x $$ D. for $$ x \leq \square $$ b. Verify that the equation is correct. $$ f(x)=\square $$, for $$ x \geq \square(x))=f(\square) $$

UpStudy AI · Accepted Answer

$$ \textbf{Step 1. Finding the Inverse Function} $$

We have
$$
f(x) = x^3 - 10.
$$
To find the inverse, start by writing
$$
y = x^3 - 10.
$$
Now swap $$x$$ and $$y$$:
$$
x = y^3 - 10.
$$
Solve for $$y$$ by adding 10 to both sides:
$$
x + 10 = y^3.
$$
Take the cube root of both sides:
$$
y = \sqrt[3]{x+10}.
$$
Thus, the inverse function is
$$
f^{-1}(x) = \sqrt[3]{x+10}.
$$

Since $$f(x)$$ is defined for all real numbers and is one-to-one, the inverse is valid for all $$x$$.

The correct choice is:  
C. $$ f^{-1}(x)=\sqrt[3]{x+10}$$, for all $$ x $$.

---

$$ \textbf{Step 2. Verifying the Inverse Function} $$

$$\underline{\text{Verification of } f(f^{-1}(x))}$$:
$$
f(f^{-1}(x)) = f\left(\sqrt[3]{x+10}\right) = \left(\sqrt[3]{x+10}\right)^3 - 10.
$$
Since the cube of a cube root returns the original argument,
$$
\left(\sqrt[3]{x+10}\right)^3 = x + 10,
$$
so
$$
f(f^{-1}(x)) = (x+10) - 10 = x.
$$

$$\underline{\text{Verification of } f^{-1}(f(x))}$$:
$$
f^{-1}(f(x)) = f^{-1}(x^3-10) = \sqrt[3]{(x^3-10)+10} = \sqrt[3]{x^3}.
$$
Again, taking the cube root of $$x^3$$ returns $$x$$, so
$$
f^{-1}(f(x)) = x.
$$

Both verifications confirm that the inverse function is correct.

---

$$ \textbf{Final Answers:} $$

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

b. We have verified that
$$
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x.
$$
a. The inverse function is $$ f^{-1}(x) = \sqrt[3]{x+10} $$ for all $$ x $$. b. We have verified that: $$ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x. $$

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