Question

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

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

y=23(x−5)
y equals Start Fraction 2 over 3 End Fraction left parenthesis x minus 5 right parenthesis

y=(3x−5)2

Answer 1

To find the inverse of the function \( g(x) = (3x + 5)^2 \), we need to follow these steps:

1. Replace \( g(x) \) with \( y \): \[ y = (3x + 5)^2 \]

2. Switch \( x \) and \( y \): \[ x = (3y + 5)^2 \]

3. Solve for \( y \):

   • Take the square root of both sides: \[ \sqrt{x} = 3y + 5 \]
   • Isolate \( 3y \): \[ 3y = \sqrt{x} - 5 \]
   • Divide by 3: \[ y = \frac{\sqrt{x} - 5}{3} \]

To find which of the given responses matches this process, we look at the options closely. None of the options shows the exact rearrangement we found, but since we want to confirm that switching variables and solving for \( y \) is reflected correctly, we determine that none of the options perfectly indicate the correct formation of the inverse.

However, the closest and most accurate option based on rearranging is: \[ y = \frac{1}{3} (2x - 5) \] which is the answer format suggesting a switch and solve for \( y \), but it’s not a direct result from the initial function provided.

If this process of rearranging and switching was the intent, the representation does seem off. Make sure to verify with the initial equation provided.