Problem 10
Question
For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$ f(x)=\frac{4}{x-1} $$
Step-by-Step Solution
Verified Answer
Domain: \((-\infty, 1) \cup (1, \infty)\); Vertical Asymptote: \(x = 1\); Horizontal Asymptote: \(y = 0\).
1Step 1: Find the Domain
The domain of a function includes all the real numbers for which the function is defined. Since the function \(f(x) = \frac{4}{x-1}\) includes a fraction, we need to ensure that the denominator is not zero. Set the denominator equal to zero and solve: \(x - 1 = 0\) which gives \(x = 1\). Therefore, the domain is all real numbers except \(x = 1\), or \((-\infty, 1) \cup (1, \infty)\).
2Step 2: Determine Vertical Asymptotes
Vertical asymptotes occur where the function is undefined and the limit of the function as it approaches that point is infinite. Since \(f(x)\) is undefined at \(x = 1\), this is also where a vertical asymptote occurs. Therefore, there is a vertical asymptote at \(x = 1\).
3Step 3: Find Horizontal Asymptotes
Horizontal asymptotes describe the behavior of \(f(x)\) as \(x\) approaches infinity or negative infinity. For \(f(x) = \frac{4}{x-1}\), as \(x\) approaches infinity or negative infinity, the term \((x - 1)\) dominates and the fraction approaches zero. Hence, the horizontal asymptote is \(y = 0\).
Key Concepts
DomainVertical AsymptotesHorizontal Asymptotes
Domain
In mathematics, the domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with rational functions, like \(f(x) = \frac{4}{x-1}\), it's crucial to ensure that the denominator never equals zero because division by zero is undefined. Thus, we need to find which values make the denominator zero and exclude them from the domain.
To find the domain:
Therefore, the domain of \(f(x) = \frac{4}{x-1}\) is all real numbers except \(x = 1\). In interval notation, this is expressed as \((-\infty, 1) \cup (1, \infty)\). Breaking it down like this helps ensure you don't accidentally use values that make the function undefined.
To find the domain:
- Set the denominator to zero: \(x - 1 = 0\).
- Solve for \(x\): \(x = 1\).
- This means the function is undefined at \(x = 1\).
Therefore, the domain of \(f(x) = \frac{4}{x-1}\) is all real numbers except \(x = 1\). In interval notation, this is expressed as \((-\infty, 1) \cup (1, \infty)\). Breaking it down like this helps ensure you don't accidentally use values that make the function undefined.
Vertical Asymptotes
Vertical asymptotes are lines that a function approaches but never touches or crosses. They occur at points where the function becomes undefined and the limits of the function as it approaches these points are infinite.
For \(f(x) = \frac{4}{x-1}\), we determined that the function becomes undefined at \(x = 1\). Here, as \(x\) gets closer to 1 from both sides, the value of \(f(x)\) increases or decreases without bound (it goes to positive or negative infinity).
Thus, there is a vertical asymptote at \(x = 1\). It's crucial to recognize vertical asymptotes because they are important in understanding the overall shape and behavior of a function. They are represented as vertical lines on the graph of the function.
For \(f(x) = \frac{4}{x-1}\), we determined that the function becomes undefined at \(x = 1\). Here, as \(x\) gets closer to 1 from both sides, the value of \(f(x)\) increases or decreases without bound (it goes to positive or negative infinity).
Thus, there is a vertical asymptote at \(x = 1\). It's crucial to recognize vertical asymptotes because they are important in understanding the overall shape and behavior of a function. They are represented as vertical lines on the graph of the function.
Horizontal Asymptotes
Horizontal asymptotes illustrate how a function behaves as \(x\) approaches very large positive or negative values. They're like a threshold the function gets closer to but never usually exceeds.
For \(f(x) = \frac{4}{x-1}\), as \(x\) approaches either infinity (positive or negative), the value of \(f(x)\) approaches zero. This happens because in the expression \(\frac{4}{x-1}\), the numerator stays constant (4), while the denominator becomes very large either positively or negatively, making the overall fraction very small and close to zero.
Therefore, the horizontal asymptote for this function is \(y = 0\). Understanding horizontal asymptotes helps you predict the end-behavior of the function, which is vital for graphing and analyzing trends accurately.
For \(f(x) = \frac{4}{x-1}\), as \(x\) approaches either infinity (positive or negative), the value of \(f(x)\) approaches zero. This happens because in the expression \(\frac{4}{x-1}\), the numerator stays constant (4), while the denominator becomes very large either positively or negatively, making the overall fraction very small and close to zero.
Therefore, the horizontal asymptote for this function is \(y = 0\). Understanding horizontal asymptotes helps you predict the end-behavior of the function, which is vital for graphing and analyzing trends accurately.
Other exercises in this chapter
Problem 10
For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as \(x\) and whe \(x=4, \quad y=2\).
View solution Problem 10
For the following exercises, find the inverse of the function on the given domain. $$ f(x)=9-x^{2},[0, \infty) $$
View solution Problem 10
For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(5 x^{5}-4 x^{4}+3 x^{3}-2 x^{2}+x-1\right) \div(x+6) $$
View solution Problem 10
For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(x^{3}-126\right) \div(x-5) $$
View solution