Question

Given g(x)=(3x+5)/2
, which of the following shows that creating the equation of the inverse means switching the variables and solving for y?(1 point)
Responses

y=(3x−5)2
y equals Start Fraction left parenthesis 3 x minus 5 right parenthesis over 2 End Fraction

y=13(2x−5)
y equals Start Fraction 1 over 3 End Fraction left parenthesis 2 x minus 5 right parenthesis

y=(2x+5)3
y equals Start Fraction left parenthesis 2 x plus 5 right parenthesis over 3 End Fraction

y=23(x−5)

Answer 1

To find the inverse of the function \( g(x) = \frac{3x + 5}{2} \), we follow these steps:

1. **Replace \( g(x) \) with \( y \):**
\[
y = \frac{3x + 5}{2}
\]

2. **Switch the variables \( x \) and \( y \):**
\[
x = \frac{3y + 5}{2}
\]

3. **Solve for \( y \):**
\[
2x = 3y + 5 \quad \text{(Multiply both sides by 2)}
\]
\[
3y = 2x - 5 \quad \text{(Subtract 5 from both sides)}
\]
\[
y = \frac{2x - 5}{3} \quad \text{(Divide both sides by 3)}
\]

Thus, the inverse function is:
\[
g^{-1}(x) = \frac{2x - 5}{3}
\]

Now, we can look at the given responses to see which one matches this result:

1. **Response**: \( y = \frac{3x - 5}{2} \)
- This does not match.

2. **Response**: \( y = \frac{1}{3}(2x - 5) \)
- This is equivalent to \( y = \frac{2x - 5}{3} \) since multiplying \( \frac{1}{3} \) by \( (2x - 5) \) gives the same expression.

3. **Response**: \( y = \frac{2x + 5}{3} \)
- This does not match.

4. **Response**: \( y = \frac{2}{3}(x - 5) \)
- This does not match.

The best response that expresses the equation of the inverse is:
**y = \frac{1}{3}(2x - 5)**.