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. (1 point) Statement # is correct.
Answers
GPT-4o mini
To determine which statement is correct regarding the inverse of the function \( g(x) = x^3 - 2 \), we first need to check if the function is one-to-one and then calculate its inverse.
1. **One-to-One Check**: A function is one-to-one if it passes the horizontal line test, meaning that for any two different inputs, the outputs should be different. The function \( g(x) = x^3 - 2 \) is a cubic function, and cubic functions are one-to-one; they are monotonically increasing for all x. This means that the function is indeed one-to-one.
2. **Finding the Inverse**: To find the inverse function, we set \( g(x) = y \):
\[
y = x^3 - 2.
\]
Now we solve for \( x \) in terms of \( y \):
\[
y + 2 = x^3 \quad \Rightarrow \quad x = (y + 2)^{1/3}.
\]
Thus, the inverse function can be expressed as:
\[
g^{-1}(y) = (y + 2)^{1/3}.
\]
If we denote the inverse function as \( g^{-1}(x) \) by replacing \( y \) with \( x \), we have:
\[
g^{-1}(x) = (x + 2)^{1/3}.
\]
Now we can evaluate the 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, as we established that the function is one-to-one).
Since none of the provided statements correctly describe the inverse of \( g(x) \), we can conclude that none of the statements are correct. However, if you are required to select a statement, please clarify or review the statements given regarding the inverse of \( g(x) \).
1. **One-to-One Check**: A function is one-to-one if it passes the horizontal line test, meaning that for any two different inputs, the outputs should be different. The function \( g(x) = x^3 - 2 \) is a cubic function, and cubic functions are one-to-one; they are monotonically increasing for all x. This means that the function is indeed one-to-one.
2. **Finding the Inverse**: To find the inverse function, we set \( g(x) = y \):
\[
y = x^3 - 2.
\]
Now we solve for \( x \) in terms of \( y \):
\[
y + 2 = x^3 \quad \Rightarrow \quad x = (y + 2)^{1/3}.
\]
Thus, the inverse function can be expressed as:
\[
g^{-1}(y) = (y + 2)^{1/3}.
\]
If we denote the inverse function as \( g^{-1}(x) \) by replacing \( y \) with \( x \), we have:
\[
g^{-1}(x) = (x + 2)^{1/3}.
\]
Now we can evaluate the 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, as we established that the function is one-to-one).
Since none of the provided statements correctly describe the inverse of \( g(x) \), we can conclude that none of the statements are correct. However, if you are required to select a statement, please clarify or review the statements given regarding the inverse of \( g(x) \).