Question

question 8 of 20
give the equation of the horizontal asymptote of the function shown below.
( g(x) = \frac{2x}{x + 5} )

a. ( x = 2 )

b. ( y = 2 )

c. ( y = \frac{2}{5} )

d. ( y = -5 )

Explanation:

Step1: Recall the rule for horizontal asymptotes of rational functions

For a rational function \( g(x)=\frac{f(x)}{h(x)} \), where \( f(x) \) and \( h(x) \) are polynomials, if the degrees of \( f(x) \) and \( h(x) \) are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The degree of a polynomial is the highest power of \( x \) in it.
In the function \( g(x)=\frac{2x}{x + 5} \), the degree of the numerator \( f(x)=2x \) is \( 1 \) (since the highest power of \( x \) is \( 1 \)) and the degree of the denominator \( h(x)=x + 5 \) is also \( 1 \) (highest power of \( x \) is \( 1 \)). So, we use the rule for equal degrees.

Step2: Find the leading coefficients

The leading coefficient of the numerator \( 2x \) is \( 2 \) (the coefficient of the highest - power term of \( x \)). The leading coefficient of the denominator \( x+5 \) is \( 1 \) (the coefficient of \( x \), which is the highest - power term of \( x \) in the denominator).

Step3: Calculate the horizontal asymptote

Using the rule for rational functions with equal degrees of numerator and denominator, the horizontal asymptote \( y \) is the ratio of the leading coefficients. So, \( y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}=\frac{2}{1} = 2 \).

Answer:

B. \( y = 2 \)