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Magazine Chapter 3: Problem 11
Which functions are even or odd or periodic? Find all asymptotes: \(y=b\) or \(x=a\) or \(y=m x+b\). Draw roughy by hand or smoothy by computer. $$ f(x)=\frac{x \sin x}{x^{2}-1} $$
Short Answer
Expert verified
The function is even, has vertical asymptotes at \(x=\pm 1\), and a horizontal asymptote at \(y=0\).
Step by step solution
01
Determine Even or Odd Function
To determine if the function is even or odd, evaluate \(f(-x)\) and compare it with \(f(x)\).- \(f(x) = \frac{x \sin x}{x^2 - 1}\)- \(f(-x) = \frac{-x \sin(-x)}{(-x)^2 - 1} = \frac{-x (-\sin x)}{x^2 - 1} = \frac{x \sin x}{x^2 - 1} = f(x)\)Since \(f(-x) = f(x)\), the function is even.
02
Check for Periodicity
A function is periodic if there is a smallest positive number \(T\) such that \(f(x+T) = f(x)\) for all \(x\). The function involves \(\sin(x)\) which is periodic with period \(2\pi\), but the rational component \(\frac{x}{x^2 - 1}\) does not have a periodic nature over the real number domain. Therefore, \(f(x)\) is not periodic.
03
Find Vertical Asymptotes
Vertical asymptotes occur where the denominator equals zero and the numerator is non-zero. Solve \(x^2 - 1 = 0\).- \(x^2 - 1 = 0 \Rightarrow x = \pm 1\)Check the numerator at \(x = \pm 1\):- \(\lim_{x \to 1} \frac{x \sin x}{x^2 - 1} = \frac{1 \times \sin 1}{0}\), indicating a vertical asymptote.- \(\lim_{x \to -1} \frac{x \sin x}{x^2 - 1} = \frac{-1 \times \sin(-1)}{0}\), also indicating a vertical asymptote.Thus, \(x = 1\) and \(x = -1\) are vertical asymptotes.
04
Check for Horizontal Asymptotes
To find horizontal asymptotes, evaluate the limits of \(f(x)\) as \(x\to \infty\) and \(x\to -\infty\).- \(\lim_{x \to \infty} \frac{x \sin x}{x^2 - 1} = \lim_{x \to \infty} \frac{\sin x}{x} = 0\)- \(\lim_{x \to -\infty} \frac{x \sin x}{x^2 - 1} = \lim_{x \to -\infty} \frac{\sin x}{x} = 0\)The horizontal asymptote is \(y = 0\).
05
Sketch the Graph
Based on the analysis, sketch the function considering:- Symmetry about the y-axis indicating it is even.- Vertical asymptotes at \(x = 1\) and \(x = -1\).- Horizontal asymptote along \(y = 0\).- The function's value approaching zero as \(x\) goes to \( \pm \infty \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
even function
An even function is a special type of function with symmetry along the y-axis. This characteristic means that whatever value you get from plugging in a positive value for point x, you will get the same for the negative value (-x). For a function to be classified as even, it must satisfy the condition: \(f(-x) = f(x)\). This type of symmetry is helpful when graphing because it shows that the graph will be a mirror image on both sides of the y-axis.
- Graphically: Symmetry along the y-axis.
- Algebraically: \(f(-x) = f(x)\).
odd function
Odd functions have a different kind of symmetry, which rotates around the origin. That means the output value for a positive input is the negative of the output value for a negative input. In mathematical terms, this is written as: \(f(-x) = -f(x)\). This property means when you plot the graph of an odd function, it will result in a symmetric plot relative to the origin (0,0).
- Graphically: Symmetric around the origin.
- Algebraically: \(f(-x) = -f(x)\).
periodic function
Periodic functions repeat their values in regular intervals. The smallest positive interval over which a function repeats itself is called the period \(T\). Trigonometric functions, such as sine and cosine, are classic examples of periodic functions, each having a period of \(2\pi\). However, not all functions containing periodic components are themselves periodic over the real numbers. In the function \(f(x) = \frac{x \sin x}{x^2 - 1}\), the sine component repeats, but its rational factor does not, making it non-periodic.
- Graphically: Repeats values over intervals.
- Example: Sine and cosine with period \(2\pi\).
- Not every component guarantees the entire function's periodicity.
vertical asymptotes
Vertical asymptotes are lines where the value of a function approaches infinity. These typically occur when the denominator of a rational function reaches zero while the numerator remains non-zero. For the function \(f(x) = \frac{x \sin x}{x^2 - 1}\), vertical asymptotes occur at points \(x = \pm 1\), because the expression \(x^2 - 1\) becomes zero while \(x \sin x\) stays non-zero.
- Indicates function value approaches infinity.
- Occurs where denominator is zero and numerator is non-zero.
- Example Here: \(x = 1\) and \(x = -1\).
horizontal asymptotes
Horizontal asymptotes are horizontal lines that a function approaches as \(x\) moves towards positive or negative infinity. These asymptotes represent the limiting behavior of a function. For the function \(f(x) = \frac{x \sin x}{x^2 - 1}\), as \(x\) approaches infinity or negative infinity, the function approaches the horizontal asymptote at \(y = 0\). This indicates that as the input values become very large, the output of the function approaches zero.
- Represents the limiting value of a function as \(x\) goes to \(\pm\infty\).
- Example Here: \(y = 0\).
