Question

shilpa’s gym membership includes a one-time fee of $20. she then pays a discounted fee of $5 for each visit. the function that shows her average cost after x visits is: f(x) = (20 + 5x)/x recall the general form of a rational function: f(x) = (a_nx^n +... + a_1x + a_0)/(b_mx^m +... + b_1x + b_0) the horizontal asymptote is y =

Explanation:

Step1: Analyze the rational function

The given function is \( f(x)=\frac{20 + 5x}{x} \), which can be rewritten as \( f(x)=\frac{5x+20}{x} \). For a rational function \( f(x)=\frac{a_nx^n+\cdots+a_1x + a_0}{b_mx^m+\cdots+b_1x + b_0} \), we compare the degrees of the numerator and the denominator. The degree of the numerator (highest power of \( x \)): in \( 5x + 20 \), the degree is \( 1 \) (since the power of \( x \) in \( 5x \) is \( 1 \)). The degree of the denominator: in \( x \), the degree is \( 1 \).

Step2: Find the horizontal asymptote

When the degrees of the numerator and the denominator are equal (both degree \( 1 \) here), the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator (coefficient of the highest power of \( x \)) is \( 5 \), and the leading coefficient of the denominator is \( 1 \). But wait, let's also simplify the function: \( f(x)=\frac{5x+20}{x}=\frac{5x}{x}+\frac{20}{x}=5+\frac{20}{x} \). As \( x \) approaches positive or negative infinity, \( \frac{20}{x} \) approaches \( 0 \). So \( f(x) \) approaches \( 5 \). Alternatively, using the degree rule: since degree of numerator (\( n = 1 \)) equals degree of denominator (\( m = 1 \)), horizontal asymptote is \( y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}=\frac{5}{1}=5 \).

Answer:

\( 5 \)