Question

Select the statement that is correct about the inverse of g(x)=x3−2
.(1 point)
Responses

Statement#1: The inverse is g−1(x)=3x+2
Statement#1: The inverse is g inverse of x is equal to 3 x plus 2

Statement #2: The inverse is g−1(x)=3x+6
Statement #2: The inverse is g inverse of x is equal to 3 x plus 6

Statement #3: The inverse is g=1(x)=−6x
Statement #3: The inverse is g raised to the is equal to 1 power of x is equal to negative 6 x

Statement #4: The inverse does not exist because the function is not one-to-one.

Answer 1

To determine the correct statement regarding the inverse of the function \( g(x) = x^3 - 2 \), let's analyze the function:

1. **One-to-One Check**: A function needs to be one-to-one (bijective) to have an inverse. The function \( g(x) = x^3 - 2 \) is a cubic function, and cubic functions are always one-to-one since they continue to increase or decrease without repeating any values. Therefore, an inverse does exist.

2. **Finding the Inverse**: To find the inverse, we will set \( y = g(x) \):
\[
y = x^3 - 2
\]

Now, solving for \( x \):
\[
y + 2 = x^3
\]
\[
x = \sqrt[3]{y + 2}
\]

Thus, the inverse function is:
\[
g^{-1}(x) = \sqrt[3]{x + 2}
\]

Now we can evaluate the statements:

- **Statement #1**: \( g^{-1}(x) = 3x + 2 \) (Incorrect)
- **Statement #2**: \( g^{-1}(x) = 3x + 6 \) (Incorrect)
- **Statement #3**: \( g^{-1}(x) = -6x \) (Incorrect)
- **Statement #4**: "The inverse does not exist because the function is not one-to-one." (Incorrect)

None of the provided statements are correct. The correct inverse function is \( g^{-1}(x) = \sqrt[3]{x + 2} \).