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Magazine Chapter 3: Problem 33
Sketch the graph of \(f\). $$f(x)=\frac{x-1}{x^{3}-4 x}$$
Short Answer
Expert verified
The graph has x-intercept at \(x = 1\), vertical asymptotes at \(x = -2, 0, 2\), and a horizontal asymptote at \(y = 0\).
Step by step solution
01
Identify the Domain
First, find the values of \(x\) for which the function is undefined. The function is undefined where the denominator is zero: \(x^3 - 4x = 0\). Factor the equation as \(x(x^2 - 4) = 0\), which gives us \(x(x - 2)(x + 2) = 0\). Thus, \(x = 0, 2, -2\) are the points where the function is undefined.
02
Find Intercepts
To find the y-intercept, set \(x = 0\): \(f(0)\) is undefined, so no y-intercept exists. To find the x-intercept, set \(f(x) = 0\) which gives \(x - 1 = 0\), hence \(x = 1\) is the x-intercept. The graph passes through the point \((1, 0)\).
03
Analyze Asymptotes
For vertical asymptotes, set the denominator equal to zero: \(x = 0, 2, -2\) due to \(x(x-2)(x+2) = 0\). For horizontal asymptotes, compare the degrees. The degree of the numerator is 1 and the denominator is 3, implying a horizontal asymptote at \(y = 0\).
04
Determine End Behavior
As \(x\) approaches infinity, the function behaves like \(\frac{1}{x^2}\), so the function approaches the horizontal asymptote \(y = 0\). As \(x\) approaches negative infinity, the function similarly approaches \(y = 0\).
05
Sketch the Graph
Based on the intercepts, asymptotes, and end behavior, sketch the function. The graph crosses the x-axis at \(x = 1\) and has vertical asymptotes at \(x = -2, 0, 2\). The graph approaches but never reaches the horizontal asymptote \(y = 0\) as \(x\) moves towards positive or negative infinity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Domain Identification
Identifying the domain of a function is a crucial step when sketching its graph. The domain tells us all the possible values of \( x \) for which the function is defined. In the given exercise, the function is expressed as a fraction:\[f(x) = \frac{x-1}{x^3 - 4x}\]The function is undefined wherever the denominator equals zero. Therefore, we set \( x^3 - 4x = 0 \) and solve for \( x \). By factoring, we find:\[x(x^2 - 4) = x(x-2)(x+2) = 0\]This implies that \( x = 0, x = 2, \) and \( x = -2 \) are points where the function is undefined. Hence, the domain of the function excludes these points. It's important to remember:
- The domain includes all real numbers except where the denominator is zero.
- Always factor completely to find all such points.
Intercepts
Finding intercepts helps determine where the graph crosses the axes. To find y-intercepts, we set \( x = 0 \). In our case, since \( f(0) \) is undefined, the function has no y-intercept. Next, finding x-intercepts requires solving the equation \( f(x) = 0 \). For our function:\[\frac{x-1}{x^3 - 4x} = 0\]The numerator must be zero, hence \( x - 1 = 0 \), giving us \( x = 1 \). Thus, the function crosses the x-axis at the point \( (1, 0) \). Keep in mind:
- If \( f(0) \) is undefined, there is no y-intercept.
- X-intercepts occur where the numerator equals zero.
Asymptotes
Understanding asymptotes is critical for sketching rational functions. Vertical asymptotes occur where the function is undefined due to a zero denominator. Our function has vertical asymptotes at \( x = 0, x = 2, \) and \( x = -2 \).For horizontal asymptotes, observe the degree of the numerator (1) and the denominator (3). Since the denominator's degree is larger, the horizontal asymptote is \( y = 0 \). Key points include:
- Vertical asymptotes are found where the denominator is zero, but the numerator is not.
- Horizontal asymptotes depend on the relative degrees of the polynomials.
End Behavior
End behavior describes how a function behaves as \( x \) approaches infinity or negative infinity. It helps predict the function's behavior at extreme values of \( x \). For the function given:As \( x \to \infty \) or \( x \to -\infty \), the term \( \frac{x-1}{x^3 - 4x} \) behaves similarly to \( \frac{1}{x^2} \), tending towards zero. Therefore, the function approaches the horizontal asymptote \( y = 0 \) on either side.Some tips about end behavior:
- Use the highest degree terms to approximate behavior at infinity.
- End behavior often mirrors the horizontal asymptote in rational functions.
