Problem 24
Question
Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$g(x)=\frac{x}{x^{2}-9}$$
Step-by-Step Solution
Verified Answer
The function \(g(x)=\frac{x}{x^{2}-9}\) has x-intercepts and a y-intercept at \(0\), vertical asymptotes at \(x=\pm3\), a horizontal asymptote at \(y=0\), and no holes. The graph can be drawn using these elements, with the graph approaching but not crossing the asymptotes.
1Step 1: Identify the function
The given rational function is \(g(x)=\frac{x}{x^{2}-9}\).
2Step 2: Find the x-intercepts and y-intercept
The x-intercepts are found by setting \(g(x)=0\) and solving for \(x\). This leads to \(x=0\). The y-intercept is found by setting \(x=0\) and solving for \(g(x)\), yielding \(g(0)=0\), so the y-intercept is also \(0\).
3Step 3: Find the domain and vertical asymptotes
The domain of a rational function is all real numbers except those for which the denominator is zero. Setting \(x^{2}-9=0\) gives \(x=\pm3\), so these are our vertical asymptotes.
4Step 4: Find the horizontal asymptote
If the degree of the denominator of a rational function is greater than the degree of the numerator, then the line \(y=0\) is a horizontal asymptote. As such, our function has a horizontal asymptote at \(y=0\).
5Step 5: Check for any holes
A hole in the graph of a rational function happens when a factor in the numerator and the denominator cancel out. However, as there are no cancellations in our function, there are no holes.
6Step 6: Graph the function
Plot the intercepts, asymptotes, and plot additional points if needed to get the overall shape of the graph. Keep in mind that the graph should approach the asymptotes but never cross them.
Key Concepts
Graph SketchingVertical AsymptotesHorizontal AsymptotesInterceptsDomain of a Function
Graph Sketching
Graph sketching for rational functions involves understanding the behavior and important features of the equation. For the function \(g(x) = \frac{x}{x^2 - 9}\), you start by identifying key features like intercepts, asymptotes, and any potential holes.
These help you determine the general shape of the graph. Once calculated, plot these elements on the graph and sketch the curve accordingly.
The graph approaches but does not cross vertical and horizontal asymptotes, except at intercepts. This visual representation helps in understanding how the function behaves for different values of \(x\).
These help you determine the general shape of the graph. Once calculated, plot these elements on the graph and sketch the curve accordingly.
The graph approaches but does not cross vertical and horizontal asymptotes, except at intercepts. This visual representation helps in understanding how the function behaves for different values of \(x\).
Vertical Asymptotes
Vertical asymptotes occur in rational functions where the denominator equals zero, causing undefined points in the function. For \(g(x) = \frac{x}{x^2 - 9}\), we identify these locations by solving \(x^2 - 9 = 0\).
This factors to \((x-3)(x+3)=0\), giving us vertical asymptotes at \(x = 3\) and \(x = -3\).
These are critical because the function approaches infinity near these points and the graph will shoot up or down, depending on the direction from which you approach the asymptote.
Remember, vertical asymptotes indicate that the function does not exist at those exact \(x\) values.
This factors to \((x-3)(x+3)=0\), giving us vertical asymptotes at \(x = 3\) and \(x = -3\).
These are critical because the function approaches infinity near these points and the graph will shoot up or down, depending on the direction from which you approach the asymptote.
Remember, vertical asymptotes indicate that the function does not exist at those exact \(x\) values.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a function as \(x\) approaches positive or negative infinity.
In \(g(x) = \frac{x}{x^2 - 9}\), the degree of the numerator (1) is less than that of the denominator (2).
This tells us that the horizontal asymptote of the function is at \(y = 0\).
While the graph may cross a horizontal asymptote, the ends of the graph will follow the direction of the asymptote. It helps provide a boundary for the behavior of the function at large values.
In \(g(x) = \frac{x}{x^2 - 9}\), the degree of the numerator (1) is less than that of the denominator (2).
This tells us that the horizontal asymptote of the function is at \(y = 0\).
While the graph may cross a horizontal asymptote, the ends of the graph will follow the direction of the asymptote. It helps provide a boundary for the behavior of the function at large values.
Intercepts
Intercepts are points where the graph crosses the x or y axis.
To find the x-intercepts of \(g(x) = \frac{x}{x^2 - 9}\), set \(g(x) = 0\), which results in \(x = 0\) as the x-intercept.
This tells us that the graph will cross the x-axis at this point.
The y-intercept is found by evaluating the function at \(x = 0\), resulting in \(g(0) = 0\).
It indicates that the graph also intersects the y-axis at the origin \((0,0)\). Gathering these intercepts is essential for accurate graph placement.
To find the x-intercepts of \(g(x) = \frac{x}{x^2 - 9}\), set \(g(x) = 0\), which results in \(x = 0\) as the x-intercept.
This tells us that the graph will cross the x-axis at this point.
The y-intercept is found by evaluating the function at \(x = 0\), resulting in \(g(0) = 0\).
It indicates that the graph also intersects the y-axis at the origin \((0,0)\). Gathering these intercepts is essential for accurate graph placement.
Domain of a Function
The domain of a function encompasses all possible x-values that can be input into the function to yield a real number.
For the function \(g(x) = \frac{x}{x^2 - 9}\), the domain excludes any x-values that make the denominator zero, which in this case are \(x = 3\) and \(x = -3\).
Therefore, the domain of the function is all real numbers except \(x = 3\) and \(x = -3\), or expressed in interval notation as \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).
Understanding the domain ensures the function is used properly within its constraints.
For the function \(g(x) = \frac{x}{x^2 - 9}\), the domain excludes any x-values that make the denominator zero, which in this case are \(x = 3\) and \(x = -3\).
Therefore, the domain of the function is all real numbers except \(x = 3\) and \(x = -3\), or expressed in interval notation as \((-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).
Understanding the domain ensures the function is used properly within its constraints.
Other exercises in this chapter
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