Question

Establish which of the following options is the inverse of f(x)=x3+3
.(1 point)
Responses

f−1(x)=x+3−−−−−√3
f inverse left parenthesis x right parenthesis equals root index 3 Start Root x plus 3 End Root

f−1(x)=x−−√3+3
f inverse left parenthesis x right parenthesis equals root index 3 Start Root x End Root plus 3

f−1(x)=x−3−−−−−√3
f inverse left parenthesis x right parenthesis equals root index 3 Start Root x minus 3 End Root

f−1(x)=x−−√3−3
f inverse left parenthesis x right parenthesis equals root index 3 Start Root x End Root minus 3

Answer 1

To find the inverse of the function \( f(x) = x^3 + 3 \), we need to follow these steps:

1. **Replace f(x) with y**:
\[ y = x^3 + 3 \]

2. **Solve for x in terms of y**:
\[ y - 3 = x^3 \]
\[ x = \sqrt[3]{y - 3} \]

3. **Replace y with f^{-1}(x)**:
\[ f^{-1}(x) = \sqrt[3]{x - 3} \]

Now we can compare this result with the given options to find the correct one.

The correct inverse function \( f^{-1}(x) \) is:

\[
f^{-1}(x) = \sqrt[3]{x - 3}
\]

Now, let’s look for that in the options:

- **Option 1**: \( \sqrt[3]{x + 3} \) → Not correct.
- **Option 2**: \( \sqrt[3]{x} + 3 \) → Not correct.
- **Option 3**: \( \sqrt[3]{x - 3} \) → This is correct.
- **Option 4**: \( \sqrt[3]{x} - 3 \) → Not correct.

Thus, the inverse function is:

**Option 3**: \( f^{-1}(x) = \sqrt[3]{x - 3} \) is the correct answer.