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Given g(x)=(3x+5)2

𝑔
(
π‘₯
)
=
(
3
π‘₯
+
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=23(xβˆ’5)
𝑦
=
2
3
(
π‘₯
βˆ’
5
)
y equals Start Fraction 2 over 3 End Fraction left parenthesis x minus 5 right parenthesis

y=(3xβˆ’5)2
𝑦
=
(
3
π‘₯
βˆ’
5
)
2
y equals Start Fraction left parenthesis 3 x minus 5 right parenthesis over 2 End Fraction

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

y=13(2xβˆ’5)

1 answer

To find the inverse of the function \( g(x) = (3x + 5)^2 \), we start by following these steps:

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

  2. Switch \( x \) and \( y \) (this step represents finding the inverse): \[ x = (3y + 5)^2 \]

  3. Solve for \( y \):

    • First, take the square root of both sides: \[ \sqrt{x} = 3y + 5 \quad \text{(Note: this assumes } x \geq 0 \text{)} \]
    • Then, isolate \( y \): \[ 3y = \sqrt{x} - 5 \] \[ y = \frac{\sqrt{x} - 5}{3} \]

Consequently, the inverse doesn’t quite match any of the choices given, but let's assess closely:

None of the options provided in your responses is written correctly as an inverse of \( g(x) \). However, based on the structure you provided, you could potentially be looking for a specific form or representation.

Among your options:

  • y=23(xβˆ’5) translates to \( y = \frac{2}{3}(x - 5) \), which doesn't align to the form we derived.
  • The other options appear misrepresented or structurally incorrect concerning the inverse.

Since none of the options strictly shows the properties of switching variables and solving for \( y \), none properly illustrates the formation of the inverse.

Based on what you've provided, the response that most closely resembles the method of switching variables (despite being incorrect) is: y=23(xβˆ’5)