Question

Select the statement that is correct about the inverse of g(x)=
3
x
​
−2.
A. The inverse is g
−1
(x)=3x+2.
B. The inverse is g
−1
(x)=3x+6.
C. The inverse is g
−1
(x)=−6x.
D. The inverse does not exist because the function is not one-to-one.

Answer 1

To find the inverse of the function \( g(x) = 3x - 2 \), we follow these steps:

1. **Replace \( g(x) \) with \( y \)**:
\[
y = 3x - 2
\]

2. **Swap \( x \) and \( y \)** (this is essentially what finding the inverse does):
\[
x = 3y - 2
\]

3. **Solve for \( y \)**:
\[
x + 2 = 3y \quad (\text{adding } 2 \text{ to both sides})
\]
\[
y = \frac{x + 2}{3} \quad (\text{dividing both sides by } 3)
\]

4. **Replace \( y \) with \( g^{-1}(x) \)**:
\[
g^{-1}(x) = \frac{x + 2}{3}
\]

Now, let's analyze the options:
A. \( g^{-1}(x) = 3x + 2 \) (incorrect)
B. \( g^{-1}(x) = 3x + 6 \) (incorrect)
C. \( g^{-1}(x) = -6x \) (incorrect)
D. The inverse does not exist because the function is not one-to-one. (incorrect; this function is one-to-one)

None of the options provided are correct for the inverse of \( g(x) = 3x - 2 \). The correct inverse we found is \( g^{-1}(x) = \frac{x + 2}{3} \).