The function $$ f(x)=(x+5)^{3} $$ 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. $$ f^{-1}(x)=\sqrt[3]{x}-5 $$, for all $$ x $$ B. $$ f^{-1}(x)= $$, for $$ x \geq $$ C. $$ f^{-1}(x)= $$, for $$ x \leq $$ D. $$ f^{-1}(x)= $$, for $$ x
eq $$ berify that the equation is correct. $$ f\left(f^{-1}(x) ight)=f(\sqrt[3]{x}-5) $$ and $$ f^{-1}(f(x))=f^{-1}(\sqrt{x} $$

Question

The function $$ f(x)=(x+5)^{3} $$ 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. $$ f^{-1}(x)=\sqrt[3]{x}-5 $$, for all $$ x $$ B. $$ f^{-1}(x)= $$, for $$ x \geq $$ C. $$ f^{-1}(x)= $$, for $$ x \leq $$ D. $$ f^{-1}(x)= $$, for $$ x 
eq $$ berify that the equation is correct. $$ f\left(f^{-1}(x)ight)=f(\sqrt[3]{x}-5) $$ and $$ f^{-1}(f(x))=f^{-1}(\sqrt{x} $$

UpStudy AI · Accepted Answer

**Step 1. Finding the Inverse Function**

We begin with the function  
$$
f(x) = (x+5)^3.
$$

To find the inverse $$ f^{-1}(x) $$, we let  
$$
y = (x+5)^3.
$$
Then, we solve for $$ x $$ in terms of $$ y $$:

1. Take the cube root of both sides:  
   $$
   \sqrt[3]{y} = x + 5.
   $$
2. Subtract 5 from both sides:  
   $$
   x = \sqrt[3]{y} - 5.
   $$

Now, interchange $$ x $$ and $$ y $$ to express the inverse function:  
$$
f^{-1}(x) = \sqrt[3]{x} - 5.
$$

**Step 2. Verifying the Inverse Function**

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

Substitute $$ f^{-1}(x) = \sqrt[3]{x} - 5 $$ into $$ f(x) $$:
$$
f\left(f^{-1}(x)\right) = f\left(\sqrt[3]{x} - 5\right) = \left[(\sqrt[3]{x} - 5) + 5\right]^3.
$$
Simplify the expression inside the parentheses:
$$
(\sqrt[3]{x} - 5) + 5 = \sqrt[3]{x}.
$$
Thus,
$$
f\left(f^{-1}(x)\right) = \left(\sqrt[3]{x}\right)^3 = x.
$$

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

Substitute $$ f(x) = (x+5)^3 $$ into $$ f^{-1}(x) $$:
$$
f^{-1}\left(f(x)\right) = f^{-1}\left((x+5)^3\right) = \sqrt[3]{(x+5)^3} - 5.
$$
Simplify the cube root:
$$
\sqrt[3]{(x+5)^3} = x + 5.
$$
Thus,
$$
f^{-1}\left(f(x)\right) = (x+5) - 5 = x.
$$

**Conclusion**

The inverse function is  
$$
f^{-1}(x)=\sqrt[3]{x}-5,
$$
and it satisfies  
$$
f\left(f^{-1}(x)\right)=x \quad \text{and} \quad f^{-1}\left(f(x)\right)=x.
$$
The inverse function is $$ f^{-1}(x) = \sqrt[3]{x} - 5 $$.

The function is one-to-one.
a. Find an equation for , the inverse function.
b. Verify that your equation is correct by showing that and .
A. , for all
B. , for
C. , for
D. , for
berify that the equation is correct.
and

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The function is one-to-one. a. Find an equation for , the inverse function. b. Verify that your equation is correct by showing that and . A. , for all B. , for C. , for D. , for berify that the equation is correct. and