To determine which function is the inverse of the given function [tex]\( f(x) = 4x^3 - 2 \)[/tex], we need to follow these steps:
### Step 1: Understand the Definition of the Inverse Function
By definition, the inverse function [tex]\( g(x) \)[/tex] of [tex]\( f(x) \)[/tex] should satisfy the condition:
[tex]\[ f(g(x)) = x \][/tex]
and
[tex]\[ g(f(x)) = x \][/tex]
### Step 2: Formally Express the Inverse Function
To find the inverse, start with:
[tex]\[ y = 4x^3 - 2 \][/tex]
Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 4y^3 - 2 \][/tex]
Now solve for [tex]\( y \)[/tex]:
[tex]\[ x + 2 = 4y^3 \][/tex]
[tex]\[ y^3 = \frac{x + 2}{4} \][/tex]
[tex]\[ y = \sqrt[3]{\frac{x + 2}{4}} \][/tex]
This means the inverse function is [tex]\( g(x) = \sqrt[3]{\frac{x + 2}{4}} \)[/tex].
### Step 3: Identify the Correct Answer from the Options
Let's check each given function:
- Option A: [tex]\( g(x)=\sqrt[3]{\frac{x-2}{4}} \)[/tex]
- Option B: [tex]\( g(x)=\sqrt[3]{\frac{x+4}{2}} \)[/tex]
- Option C: [tex]\( g(x)=\sqrt[3]{\frac{4}{x+2}} \)[/tex]
- Option D: [tex]\( g(x)=\sqrt[3]{\frac{x+2}{4}} \)[/tex]
Comparing these with our derived inverse function [tex]\( g(x) = \sqrt[3]{\frac{x + 2}{4}} \)[/tex], we see that:
- Option D ([tex]\(g(x)=\sqrt[3]{\frac{x+2}{4}} \)[/tex]) perfectly matches the derived inverse function.
### Step 4: Verification
Let’s verify Option D to be sure it's the correct inverse:
Check [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = 4 \left(\sqrt[3]{\frac{x+2}{4}}\right)^3 - 2 \][/tex]
[tex]\[ = 4 \left(\frac{x+2}{4}\right) - 2 \][/tex]
[tex]\[ = (x+2) - 2 \][/tex]
[tex]\[ = x \][/tex]
Check [tex]\( g(f(x)) \)[/tex]:
[tex]\[ g(f(x)) = \sqrt[3]{\frac{4x^3 - 2 + 2}{4}} \][/tex]
[tex]\[ = \sqrt[3]{x^3} \][/tex]
[tex]\[ = x \][/tex]
Since both conditions are satisfied, Option D is indeed the correct inverse.
### Conclusion
The function that is the inverse of [tex]\( f(x) = 4x^3 - 2 \)[/tex] is:
[tex]\[ \boxed{g(x) = \sqrt[3]{\frac{x+2}{4}}} \][/tex]
Or, referring to the options, it's Option D. Hence, the correct answer is [tex]\( \boxed{0} \)[/tex] indicating that Option D is indeed the correct inverse function.
