Question

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

Statement #1: The inverse is g−1(x)=3x+2
.

Statement #2: The inverse is g−1(x)=3x+6
.

Statement #3: The inverse is g−1(x)=−6x
.

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

Answer 1

To find the inverse of the function \( g(x) = x^3 - 2 \), we first need to determine if the function is one-to-one. A function is one-to-one if it passes the horizontal line test, meaning that for any two different \( x \) values, the output \( g(x) \) is also different.

The function \( g(x) = x^3 - 2 \) is a cubic function, and cubic functions are one-to-one since they are strictly increasing (the derivative \( g'(x) = 3x^2 \) is always non-negative).

Since \( g(x) \) is one-to-one, we can find its inverse. To find the inverse, we set \( y = g(x) \):

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

Now, let's evaluate the given statements:

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

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