Select the statement that is correct about the inverse of g(x)=x/3−2

Question

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Statement #1: The inverse is g−1(x)=3x+2
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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.

Answer 1

To find the inverse of the function \( g(x) = \frac{x}{3} - 2 \), we first rewrite the equation in terms of \( y \):

1. \( y = \frac{x}{3} - 2 \)

Next, we solve for \( x \) in terms of \( y \):

1. Add 2 to both sides:
   \( y + 2 = \frac{x}{3} \)

2. Multiply both sides by 3:
   \( x = 3(y + 2) \)

Now, we express \( x \) in terms of \( y \):

1. \( x = 3y + 6 \)

Then, we express the inverse function \( g^{-1}(x) \):

1. \( g^{-1}(x) = 3x + 6 \)

Now, we can analyze the statements:

• Statement #1: The inverse is \( g^{-1}(x) = 3x + 2 \)

  • Incorrect.

• Statement #2: The inverse is \( g^{-1}(x) = 3x + 6 \)

  • Correct.

• 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. The function \( g(x) = \frac{x}{3} - 2 \) is a linear function and is one-to-one.

The correct statement is Statement #2: The inverse is \( g^{-1}(x) = 3x + 6 \).