Question

graph the rational function.

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

start by drawing the vertical and horizontal asymptotes. then plot two points on each piece of the graph. finally, click on the graph-a-function button.

Explanation:

Find the asymptotes

Using the Domain of a Function knowledge point, we find where the denominator is zero.
The vertical asymptote occurs where \(x + 3 = 0\), which gives:

$$x = -3$$

The horizontal asymptote is found by comparing the degrees of the numerator and denominator. Since both have degree 1, we divide the leading coefficients:

$$y = \frac{2}{1} = 2$$

Plot points to the left of the vertical asymptote

We choose two \(x\)-values less than \(-3\), such as \(x = -4\) and \(x = -5\).
For \(x = -4\):

$$f(-4) = \frac{2(-4) + 3}{-4 + 3} = \frac{-5}{-1} = 5 \implies (-4, 5)$$

For \(x = -5\):

$$f(-5) = \frac{2(-5) + 3}{-5 + 3} = \frac{-7}{-2} = 3.5 \implies (-5, 3.5)$$

Plot points to the right of the vertical asymptote

We choose two \(x\)-values greater than \(-3\), such as \(x = -2\) and \(x = -1\).
For \(x = -2\):

$$f(-2) = \frac{2(-2) + 3}{-2 + 3} = \frac{-1}{1} = -1 \implies (-2, -1)$$

For \(x = -1\):

$$f(-1) = \frac{2(-1) + 3}{-1 + 3} = \frac{1}{2} = 0.5 \implies (-1, 0.5)$$

Answer:

To graph the rational function \(f(x) = \frac{2x + 3}{x + 3}\):

1. Asymptotes:

• Vertical Asymptote: \(x = -3\)
• Horizontal Asymptote: \(y = 2\)

1. Points on the left piece (\(x < -3\)):

• \((-4, 5)\)
• \((-5, 3.5)\)

1. Points on the right piece (\(x > -3\)):

• \((-2, -1)\)
• \((-1, 0.5)\)