Problem 32
Question
Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{1}{x+3}$$
Step-by-Step Solution
Verified Answer
The function \(f(x)=\frac{1}{x+3}\) has a y-intercept at (0,1/3), no x-intercepts, vertical asymptote at \(x = -3\), and horizontal asymptote at \(y = 0\). The graph is a downward-opening hyperbola with a center shifted 3 units to the left.
1Step 1: Identify and plot the intercepts
The intercepts are found where \(f(x)=0\) and where \(x=0\). For \(f(x)=0\), there are no solutions as there's no real value of \(x\) such that \(\frac{1}{x+3}=0\). Thus, there are no x-intercepts. The y-intercept is found by setting \(x=0\), yielding \(f(0)=\frac{1}{0+3} = \frac{1}{3}\), which means the y-intercept is at (0,1/3). Plot this point.
2Step 2: Identify and plot the asymptotes
The vertical asymptote is found at the value of \(x\) that makes the denominator 0 and the function undefined. In this case, that's \(x = -3\). For the horizontal asymptote, as the degree of the numerator is less than the denominator, the function approaches 0 as \(x\) approaches \(\pm\infty\). Thus, the horizontal asymptote is \(y = 0\).
3Step 3: Sketch the graph
Draw the graph so it approaches but does not cross the asymptotes. As \(x\) approaches -3 from the left, \(f(x)\) goes to \(-\infty\), and as \(x\) approaches -3 from the right, \(f(x)\) goes to \(\infty\). The function approaches \(y = 0\) as \(x\) approaches \(\pm\infty\).
Key Concepts
Vertical AsymptoteHorizontal AsymptoteFunction Intercepts
Vertical Asymptote
When graphing rational functions, vertical asymptotes are crucial to understand. They occur where the function is undefined due to the denominator being zero. Imagine a line that the function gets infinitely close to as the input heads towards a certain value, but the function never actually crosses this line. In the case of our example function, \( f(x) = \frac{1}{x+3} \), the vertical asymptote is at \( x = -3 \) because that's the value of \( x \) which makes the denominator of our fraction zero, hence the function is undefined.
Why are vertical asymptotes important? They help in predicting the behavior of the graph. As the value of \( x \) approaches the vertical asymptote, the value of the function grows without bounds, either towards positive or negative infinity depending on the direction from which it approaches. In practical terms, this means that vertical asymptotes signal where there are breaks in the graph, serving as boundaries that the graph will approach but never cross.
Why are vertical asymptotes important? They help in predicting the behavior of the graph. As the value of \( x \) approaches the vertical asymptote, the value of the function grows without bounds, either towards positive or negative infinity depending on the direction from which it approaches. In practical terms, this means that vertical asymptotes signal where there are breaks in the graph, serving as boundaries that the graph will approach but never cross.
Horizontal Asymptote
A horizontal asymptote on the other hand, refers to a line that the function approaches as the input either increases or decreases without bound. In our example, because the degree (highest exponent) of the numerator of \( f(x) = \frac{1}{x+3} \) is zero (since there is just a constant '1') and the degree of the denominator is one (since there is an \( x \) to the first power), the horizontal asymptote is at \( y = 0 \).
What's the intuition behind this? It comes down to the comparison of growth rates between the numerator and the denominator. If the numerator grows slower than the denominator, as \( x \) gets larger and larger in magnitude, the value of the function gets closer and closer to zero. Therefore, for large absolute values of \( x \), the graph will appear to settle towards the line \( y = 0 \), creating a horizontal asymptote. It acts as a horizontal boundary line that tells us the end behavior of the graph at its extremes to either side of the x-axis.
What's the intuition behind this? It comes down to the comparison of growth rates between the numerator and the denominator. If the numerator grows slower than the denominator, as \( x \) gets larger and larger in magnitude, the value of the function gets closer and closer to zero. Therefore, for large absolute values of \( x \), the graph will appear to settle towards the line \( y = 0 \), creating a horizontal asymptote. It acts as a horizontal boundary line that tells us the end behavior of the graph at its extremes to either side of the x-axis.
Function Intercepts
The intercepts of a function represent the points at which the graph crosses the axes. These are particularly helpful for initially plotting the graph of the function. There are two kinds of intercepts: x-intercepts and y-intercepts.
The x-intercepts are found by setting the output \( f(x) \) to zero and solving for \( x \). In the case of our example, no x-intercepts exist because there are no real values of \( x \) that would make \( \frac{1}{x+3} = 0 \). This means the graph will never touch the x-axis. On the contrary, the y-intercept is found by setting \( x = 0 \) in the function and finding the corresponding \( y \) value. For \( f(x) = \frac{1}{x+3} \), setting \( x = 0 \) gives us the y-intercept of \( \frac{1}{3} \), and we plot this point at \( (0, \frac{1}{3}) \) on the graph.
Understanding intercepts is key to sketching a rough outline of the function before considering more intricate details of the graph, like its asymptotic behavior, and ultimately provides the starting points from which the function extends towards its limits as defined by the asymptotes.
The x-intercepts are found by setting the output \( f(x) \) to zero and solving for \( x \). In the case of our example, no x-intercepts exist because there are no real values of \( x \) that would make \( \frac{1}{x+3} = 0 \). This means the graph will never touch the x-axis. On the contrary, the y-intercept is found by setting \( x = 0 \) in the function and finding the corresponding \( y \) value. For \( f(x) = \frac{1}{x+3} \), setting \( x = 0 \) gives us the y-intercept of \( \frac{1}{3} \), and we plot this point at \( (0, \frac{1}{3}) \) on the graph.
Understanding intercepts is key to sketching a rough outline of the function before considering more intricate details of the graph, like its asymptotic behavior, and ultimately provides the starting points from which the function extends towards its limits as defined by the asymptotes.
Other exercises in this chapter
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