Sanders Salinas
10/30/2023 · High School
\( 1 \leftarrow \quad \) The function \( f(x)=x^{3}-4 \) is one-to-one. a. Find an equation for \( f^{-1} \), the inverse function. b. Verify that your equation is correct by showing that \( f\left(f^{-1}(x)\right)=x \) and \( f^{-1}(f(x))=x \). b. Verify that the equation is correct. \( f\left(f^{-1}(x)\right)=f(\square) \) \( =\square \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = x^3 - 4 \), we will follow these steps:
### Part a: Finding the Inverse Function
1. **Set \( y = f(x) \)**:
\[
y = x^3 - 4
\]
2. **Solve for \( x \)** in terms of \( y \):
\[
y + 4 = x^3
\]
\[
x = \sqrt[3]{y + 4}
\]
3. **Express the inverse function**:
\[
f^{-1}(x) = \sqrt[3]{x + 4}
\]
### Part b: Verification
We need to verify that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
1. **Verify \( f(f^{-1}(x)) = x \)**:
\[
f(f^{-1}(x)) = f\left(\sqrt[3]{x + 4}\right)
\]
Substitute \( \sqrt[3]{x + 4} \) into \( f(x) \):
\[
= \left(\sqrt[3]{x + 4}\right)^3 - 4
\]
\[
= (x + 4) - 4
\]
\[
= x
\]
2. **Verify \( f^{-1}(f(x)) = x \)**:
\[
f^{-1}(f(x)) = f^{-1}(x^3 - 4)
\]
Substitute \( x^3 - 4 \) into \( f^{-1}(x) \):
\[
= \sqrt[3]{(x^3 - 4) + 4}
\]
\[
= \sqrt[3]{x^3}
\]
\[
= x
\]
### Conclusion
Both verifications confirm that the inverse function is correct. Thus, we have:
- The inverse function is:
\[
f^{-1}(x) = \sqrt[3]{x + 4}
\]
- Verification results:
\[
f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
\]
Quick Answer
The inverse function \( f^{-1}(x) \) is \( \sqrt[3]{x + 4} \). Verification confirms that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
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