Sure, let's carefully determine the inverse of the given function step by step.
Given the function:
[tex]\[ f(x) = \sqrt[3]{8x + 4} \][/tex]
Step 1: Set \( f(x) \) to \( y \):
[tex]\[ y = \sqrt[3]{8x + 4} \][/tex]
Step 2: Switch \( x \) and \( y \):
[tex]\[ x = \sqrt[3]{8y + 4} \][/tex]
Step 3: Solve for \( y \):
- Cube both sides to remove the cube root:
[tex]\[ x^3 = 8y + 4 \][/tex]
- Subtract 4 from both sides to isolate the term with \( y \):
[tex]\[ x^3 - 4 = 8y \][/tex]
- Divide by 8 to solve for \( y \):
[tex]\[ y = \frac{x^3 - 4}{8} \][/tex]
The inverse function \( f^{-1}(x) \) is therefore:
[tex]\[ f^{-1}(x) = \frac{x^3 - 4}{8} \][/tex]
Now we can evaluate both \( f(0) \) and \( f^{-1}(0) \).
1. Evaluate \( f(0) \):
[tex]\[ f(0) = \sqrt[3]{8 \cdot 0 + 4} = \sqrt[3]{4} \approx 1.587 \][/tex]
2. Evaluate \( f^{-1}(0) \):
[tex]\[ f^{-1}(0) = \frac{0^3 - 4}{8} = \frac{-4}{8} = -0.5 \][/tex]
Thus, the results are [tex]\( f(0) \approx 1.587 \)[/tex] and [tex]\( f^{-1}(0) = -0.5 \)[/tex].
