Problem 56 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{6-2 x}{x+3} $$

Step-by-Step Solution

Answer

The domain is all real numbers except \(-3\). The vertical asymptote is \(x = -3\); the horizontal asymptote is \(y = -2\). The graph approaches but doesn't cross these asymptotes.

Step 1: Finding the Domain

The domain of a rational function is all real numbers except those that make the denominator zero. For the function \( f(x) = \frac{6-2x}{x+3} \), set the denominator, \( x + 3 \), equal to zero and solve for \( x \). \( x + 3 = 0 \Rightarrow x = -3 \). Therefore, the domain of \( f \) is all real numbers except \( x = -3 \).

Step 2: Graphing the Function

To graph \( f(x) = \frac{6-2x}{x+3} \), use a graphing tool or software. Choose a viewing rectangle that includes the point where \( x = -3 \) to observe the vertical asymptote and extended views to see the behavior of the graph off towards positive and negative infinity.

Step 3: Finding Asymptotes

To find vertical asymptotes, set the denominator equal to zero: \( x + 3 = 0 \). Thus, there is a vertical asymptote at \( x = -3 \). For horizontal asymptotes, compare the degrees of the numerator and denominator. Both the numerator, \( -2x + 6 \), and the denominator, \( x + 3 \), have degree 1. The horizontal asymptote is given by the ratio of the leading coefficients, \( -2/1 \), which is \( y = -2 \).

Step 4: Sketching the Graph

Sketch \( f(x) = \frac{6-2x}{x+3} \) incorporating the asymptotes. Mark the vertical asymptote at \( x = -3 \) as a dashed vertical line and the horizontal asymptote at \( y = -2 \) as a dashed horizontal line. Draw the curve of the graph approaching these lines but never crossing them, showing the behavior of the function as it extends towards infinity.

Key Concepts

Domain of a Function

When dealing with a rational function like \( f(x) = \frac{6-2x}{x+3} \), finding the domain is an important first step. The domain involves determining which values of \( x \) make the function undefined. Typically, with rational functions, this occurs wherever the denominator is zero, because division by zero is undefined.

For the function given, this means we need to solve the equation \( x + 3 = 0 \). Solving gives us \( x = -3 \). Hence, the domain for \( f(x) \) is all real numbers except \( x = -3 \). You can think of the domain as the set of "legal" \( x \) values that you can input into \( f(x) \) without causing any mathematical mishaps.

• Check the denominator for zero.
• Exclude these \( x \) values from the domain.
• Write the domain using interval notation.

In this case, the domain of \( f \) is \( (-\infty, -3) \cup (-3, \infty) \).

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. For rational functions, these lines can be either vertical or horizontal.

**Vertical Asymptotes**
To identify vertical asymptotes, we look again to the denominator. Since the denominator cannot be zero, solve \( x + 3 = 0 \) to find \( x = -3 \). This tells us there is a vertical asymptote at \( x = -3 \). This line represents a point of discontinuity in the graph where the function heads towards infinity.

**Horizontal Asymptotes**
Horizontal asymptotes describe the behavior of a function as \( x \) approaches positive or negative infinity. For a function of the form \( \frac{ax + b}{cx + d} \), compare the degrees of the numerator and denominator:

• If they are the same, the horizontal asymptote is \( y = \frac{a}{c} \).

In our function, both the numerator and the denominator are degree 1. Thus, the horizontal asymptote is \( y = -2 \). This line tells us how the function behaves far out to the left and right.

Graphing Functions

Graphing a rational function like \( f(x) = \frac{6-2x}{x+3} \) involves considering both the domain and the asymptotes. Begin by marking the critical lines, typically vertical and horizontal asymptotes, on your graph. This helps guide the drawing of the function's curve.

**Plotting Asymptotes**
1. Draw a dashed vertical line at \( x = -3 \) for the vertical asymptote.
2. Draw a dashed horizontal line at \( y = -2 \) for the horizontal asymptote.

**Sketching the Curve**
The graph of \( f(x) \) will come close to these dashed lines but not touch them. As \( x \) approaches \( -3 \), either from the left or right, the curve will go towards positive or negative infinity. As \( x \) moves further from the vertical asymptote, the curve will tend to follow the horizontal asymptote, \( y = -2 \).

When the graph is sketched with the asymptotes considered, it provides a clearer picture of how the function behaves across its domain. The use of graphing software can offer additional insight by revealing more points along \( f(x) \), which help complete your sketch.