The function $$ f(x)=(x+1)^{3} $$ is one-to-one. a. Find an equation for $$ f^{-1}(x) $$, 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. A. 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 $$ x \leq \square $$ D. for all $$ x $$ $$ f^{-1}(x)=\square $$, for $$ x \geq \square $$

Question

The function $$ f(x)=(x+1)^{3} $$ is one-to-one. a. Find an equation for $$ f^{-1}(x) $$, 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. A. 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 $$ x \leq \square $$ D. for all $$ x $$ $$ f^{-1}(x)=\square $$, for $$ x \geq \square $$

UpStudy AI · Accepted Answer

a. To find the inverse function, start with

$$
y = (x+1)^3.
$$

Replace $$ y $$ with $$ x $$ and $$ x $$ with $$ y $$ to solve for $$ y $$:

$$
x = (y+1)^3.
$$

Taking the cube root of both sides gives

$$
\sqrt[3]{x} = y + 1.
$$

Subtract 1 from both sides:

$$
y = \sqrt[3]{x} - 1.
$$

Thus, the inverse function is

$$
f^{-1}(x) = \sqrt[3]{x} - 1 \quad \text{for all } x.
$$

b. Verification:

1. Compute $$ f\left(f^{-1}(x)\right) $$:

$$
f\left(f^{-1}(x)\right) = f\left(\sqrt[3]{x} - 1\right) = \left[\left(\sqrt[3]{x} - 1\right) + 1\right]^3 = \left(\sqrt[3]{x}\right)^3 = x.
$$

2. Compute $$ f^{-1}(f(x)) $$:

$$
f^{-1}(f(x)) = f^{-1}\left((x+1)^3\right) = \sqrt[3]{(x+1)^3} - 1 = (x+1) - 1 = x.
$$

Thus, the inverse function satisfies

$$
f\left(f^{-1}(x)\right) = x \quad \text{and} \quad f^{-1}\left(f(x)\right) = x.
$$

The correct choice is:

D. for all $$ x $$:  
$$ f^{-1}(x) = \sqrt[3]{x} - 1 $$, for $$ x \geq \text{ (all real numbers)} $$.
a. The inverse function is $$ f^{-1}(x) = \sqrt[3]{x} - 1 $$ for all real numbers $$ x $$. b. Verification: - $$ f\left(f^{-1}(x)\right) = x $$ - $$ f^{-1}\left(f(x)\right) = x $$ The correct choice is: D. for all $$ x $$: $$ f^{-1}(x) = \sqrt[3]{x} - 1 $$, for all real numbers $$ x $$.

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