Question

When the graph of the function f(x)=1x
is replaced by the function g(x)=f(x)+d
, its new horizontal asymptote is at y=13
. Which of the following represents the function g(x)
?(1 point)
Responses

g(x)=1x−13
g left parenthesis x right parenthesis equals Start Fraction 1 over x End Fraction minus 13

g(x)=1x+13
g left parenthesis x right parenthesis equals Start Fraction 1 over x plus 13 End Fraction

g(x)=1x+13
g left parenthesis x right parenthesis equals Start Fraction 1 over x End Fraction plus 13

g(x)=1x−13
g left parenthesis x right parenthesis equals Start Fraction 1 over x minus 13 End Fraction

Answer 1

To determine the correct form of the function \( g(x) \), let's analyze the situation given.

The original function \( f(x) = \frac{1}{x} \) has a horizontal asymptote at \( y = 0 \) as \( x \to \pm \infty \). When we modify this function to introduce \( d \) (i.e., \( g(x) = f(x) + d \)), the horizontal asymptote shifts up or down depending on the value of \( d \).

Given that the new horizontal asymptote is at \( y = 13 \), we can deduce that \( d \) must be equal to 13, since adding 13 to the original horizontal asymptote \( y = 0 \) results in \( y = 13 \).

Thus, we can express \( g(x) \) as:
\[
g(x) = f(x) + 13 = \frac{1}{x} + 13
\]

So, the correct representation of the function \( g(x) \) from the options provided is:
\[
g(x) = \frac{1}{x} + 13
\]

Therefore, the correct response is:
**g(x) = \frac{1}{x} + 13**