Questions: Identify any horizontal and vertical asymptotes and any holes in the graph. Remember to write HA and V/ as EQUATIONS. f(x) = 5x / (2x + 3)

Transcript text: Identify any horizontal and vertical asymptotes and any holes in the graph. Remember to write HA and V/ as EQUATIONS. \[ f(x)=\frac{5 x}{2 x+3} \]

Solution

Solution Steps

Step 1: Determine the Degrees of the Numerator and Denominator

The function is given by

\[ f(x) = \frac{5x}{2x + 3} \]

The degree of the numerator \(5x\) is \(1\) and the degree of the denominator \(2x + 3\) is also \(1\). Since the degrees are equal, we can find the horizontal asymptote.

Step 2: Find the Horizontal Asymptote

For rational functions where the degrees of the numerator and denominator are the same, the horizontal asymptote (HA) is given by the ratio of the leading coefficients. The leading coefficient of the numerator is \(5\) and that of the denominator is \(2\). Therefore, the horizontal asymptote is:

\[ y = \frac{5}{2} \]

Step 3: Find the Vertical Asymptote

To find the vertical asymptote (VA), we set the denominator equal to zero:

\[ 2x + 3 = 0 \]

Solving for \(x\) gives:

\[ x = -\frac{3}{2} \]

Step 4: Check for Holes

Holes occur when there are common factors in the numerator and denominator that cancel out. In this case, the numerator \(5x\) factors to \(5 \cdot x\) and the denominator \(2x + 3\) does not share any common factors with the numerator. Therefore, there are no holes in the function.