Problem 10
Question
Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{2}{x^{2}-9}$$
Step-by-Step Solution
Verified Answer
The domain of the function is \(x\in(-\infty, -3) \cup (-3, 3) \cup (3, \infty)\), the vertical asymptotes are \(x = -3\) and \(x = 3\), and the horizontal asymptote is \(y = 0\).
1Step 1: Finding the Domain
The denominator should not be zero because division by zero is undefined in mathematics. Therefore, the values of \(x^2-9\) should not be zero. Solve the equation for \(x^2-9=0\) and exclude these values from all real numbers as the domain. The solutions are \(x=3, -3\). Then the domain of this function is \(x\in(-\infty, -3) \cup (-3, 3) \cup (3, \infty)\).
2Step 2: Finding the Vertical Asymptotes
A vertical asymptote occurs when the denominator equates to zero. From the previous step, we know that the denominator equals zero when \(x = -3\) and \(x = 3\). So, the vertical asymptotes are \(x = -3\) and \(x = 3\).
3Step 3: Finding the Horizontal Asymptotes
When the degree of the denominator's polynomial is larger than the degree of the numerator's polynomial, the x-axis (\(y = 0\)) is the horizontal asymptote. In the function \(f(x)=\frac{2}{x^{2}-9}\), the degree of the denominator's polynomial (\(x^{2}-9\)) is 2, and the degree of the numerator's polynomial (2) is 0. Since 2 (the denominator's degree) is greater than 0 (the numerator's degree), \(y = 0\) is the horizontal asymptote.
Key Concepts
Domain of a FunctionVertical AsymptotesHorizontal Asymptotes
Domain of a Function
The domain of a rational function refers to all possible input values (x-values) that the function can accept without causing any undefined behavior, like division by zero. For the function \(f(x)=\frac{2}{x^{2}-9}\), the denominator is \(x^2-9\). This expression is undefined when it equals zero, as division by zero isn't allowed in mathematics.
Thus, to find the domain, we solve the equation \(x^2-9=0\). By solving this, we find \(x=3\) and \(x=-3\). These are the x-values that make the denominator zero.
Therefore, to define the domain, we exclude \(-3\) and \(3\), and write it as all real numbers except these two:
Thus, to find the domain, we solve the equation \(x^2-9=0\). By solving this, we find \(x=3\) and \(x=-3\). These are the x-values that make the denominator zero.
Therefore, to define the domain, we exclude \(-3\) and \(3\), and write it as all real numbers except these two:
- From negative infinity to -3
- Union of -3 to 3
- Union of 3 to positive infinity
Vertical Asymptotes
Vertical asymptotes are vertical lines that the graph of a function approaches but never touches. These occur where the function, particularly in rational functions, becomes undefined. For \(f(x)=\frac{2}{x^{2}-9}\), the potential vertical asymptotes occur at the same points as the zeros of the denominator.
Since we previously found that \(x = 3\) and \(x = -3\) are where the denominator equals zero, there are vertical asymptotes at these x-values. Every time the denominator of a rational function becomes zero at a specific point and doesn't cancel out with the numerator, there is a vertical asymptote there.
In conclusion, the vertical asymptotes for this function are at \(x = -3\) and \(x = 3\).
Graphically, the function will get extremely close to these lines but won't cross or touch them. This characteristic is a key element in understanding the behavior of rational functions.
Since we previously found that \(x = 3\) and \(x = -3\) are where the denominator equals zero, there are vertical asymptotes at these x-values. Every time the denominator of a rational function becomes zero at a specific point and doesn't cancel out with the numerator, there is a vertical asymptote there.
In conclusion, the vertical asymptotes for this function are at \(x = -3\) and \(x = 3\).
Graphically, the function will get extremely close to these lines but won't cross or touch them. This characteristic is a key element in understanding the behavior of rational functions.
Horizontal Asymptotes
Horizontal asymptotes describe the end behavior of a function. They indicate the y-value that the function approaches as \(x\) moves towards either positive or negative infinity. For the given function \(f(x)=\frac{2}{x^{2}-9}\), the degrees of the numerator and the denominator are integral in determining the horizontal asymptote.
The degree of the polynomial in the numerator is 0 (constant), while the degree of the polynomial in the denominator is 2. When the degree in the denominator is greater than that in the numerator:
Thus, the horizontal asymptote of this function is the x-axis, or \(y = 0\).
Understanding the horizontal asymptote helps to grasp how the function behaves at its extremes and confirms that it flattens out along the line \(y = 0\) as \(x\) approaches infinity in either direction.
The degree of the polynomial in the numerator is 0 (constant), while the degree of the polynomial in the denominator is 2. When the degree in the denominator is greater than that in the numerator:
- the horizontal asymptote is \(y = 0\)
Thus, the horizontal asymptote of this function is the x-axis, or \(y = 0\).
Understanding the horizontal asymptote helps to grasp how the function behaves at its extremes and confirms that it flattens out along the line \(y = 0\) as \(x\) approaches infinity in either direction.
Other exercises in this chapter
Problem 10
For each polynomial function, list the zeros of the polynomial and state the multiplicity of each zero. $$h(x)=(x-\sqrt{2})^{13}(x+\sqrt{2})^{7}$$
View solution Problem 10
For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$(x)=x^{3}+2 x^{2}-2 x-4 ; x=\sqrt{2},-\sqrt{3}$$
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Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$-x^{3}+x ; x-5$$
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Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touc
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