The function f of x equals Natural Log left-parenthesis x right-parenthesis EndLog has been transformed so that there is now a vertical asymptote at x equals negative 3.

Question

What is the equation of the resulting function, g of x?

Answer options with 4 options
A.
g of x equals Natural Log left-parenthesis x right-parenthesis EndLog minus 3

B.
g of x equals Natural Log left-parenthesis x right-parenthesis EndLog plus 3

C.
g of x equals Natural Log left-parenthesis x minus 3 right-parenthesis EndLog

D.
g of x equals Natural Log left-parenthesis x plus 3 right-parenthesis EndLog

Answer 1

To achieve a vertical asymptote at \( x = -3 \) using the natural logarithm function \( f(x) = \ln(x) \), we need to adjust the argument of the logarithm.

The logarithm function \( \ln(x) \) has a vertical asymptote at \( x = 0 \). To shift this vertical asymptote to \( x = -3 \), we must adjust the input to the logarithm function such that it becomes zero when \( x = -3 \).

This can be done by using the expression \( x + 3 \) inside the logarithm:

\[ g(x) = \ln(x + 3) \]

With this transformation, the function \( g(x) \) approaches negative infinity as \( x \) approaches \(-3\) from the right, which creates the desired vertical asymptote.

Thus, the correct answer is:

D. \( g(x) = \ln(x + 3) \).