Question

1. find the horizontal asymptote(s) of each rational function. a) $f(x) = \frac{3x - 15}{2x + 12}$

Explanation:

Step1: Recall Horizontal Asymptote Rule

For a rational function \( f(x) = \frac{N(x)}{D(x)} \), where \( N(x) \) and \( D(x) \) are polynomials:

• If the degree of \( N(x) \) (\( n \)) equals the degree of \( D(x) \) (\( m \)), the horizontal asymptote is \( y = \frac{\text{leading coefficient of } N(x)}{\text{leading coefficient of } D(x)} \).
• If \( n < m \), horizontal asymptote is \( y = 0 \).
• If \( n > m \), no horizontal asymptote (oblique/slant asymptote may exist).

Step2: Identify Degrees and Coefficients

For \( f(x) = \frac{3x - 15}{2x + 12} \):

• Degree of \( N(x) = 3x - 15 \) is \( 1 \) (highest power of \( x \)).
• Degree of \( D(x) = 2x + 12 \) is \( 1 \).
• Leading coefficient of \( N(x) \) is \( 3 \).
• Leading coefficient of \( D(x) \) is \( 2 \).

Step3: Apply the Rule

Since degrees are equal (\( n = m = 1 \)), horizontal asymptote is \( y = \frac{3}{2} \).

Answer:

The horizontal asymptote of \( f(x) = \frac{3x - 15}{2x + 12} \) is \( y = \frac{3}{2} \).