Question

1. what is the horizontal asymptote of $y = \frac{4x + 1}{2x}$

show your work

1. what is the excluded value of $y = \frac{4x + 1}{2x}$

show your work

Explanation:

Question 5

Step1: Analyze the rational function

For a rational function \( y=\frac{f(x)}{g(x)} \), to find the horizontal asymptote, we compare the degrees of the numerator and the denominator. Here, the numerator \( 4x + 1 \) and the denominator \( 2x \) both have degree 1 (the highest power of \( x \) is 1).

Step2: Apply the rule for horizontal asymptote (same degree)

When the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 2. So the horizontal asymptote is \( y=\frac{4}{2}=2 \).

Step1: Recall the definition of excluded value

The excluded value of a rational function \( y=\frac{f(x)}{g(x)} \) is the value of \( x \) that makes the denominator \( g(x)=0 \) (since division by zero is undefined).

Step2: Solve for \( x \) when the denominator is zero

For the function \( y = \frac{4x + 1}{2x} \), set the denominator \( 2x=0 \). Solving this equation gives \( x = 0 \) (divide both sides by 2).

Answer:

\( y = 2 \)

Question 6