Problem 280 - Calculus Volume 1

Question

Find the horizontal and vertical asymptotes. . \(f(x)=\frac{1}{x-1}-2 x\)

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 1 \). No horizontal asymptote.

Step 1: Identify Vertical Asymptotes

To find vertical asymptotes, we need to determine where the function is undefined. The function is undefined where the denominator of any fraction is zero. For this function, the fraction part is \( \frac{1}{x-1} \). Set the denominator equal to zero: \( x-1 = 0 \). Solve for \( x \): \( x = 1 \). Hence, there is a vertical asymptote at \( x = 1 \).

Step 2: Identify Horizontal Asymptotes

To find horizontal asymptotes, examine the behavior of \( f(x) \) as \( x \) approaches infinity. The function can be written as \( f(x) = \frac{1}{x-1} - 2x \). As \( x \) approaches infinity, \( \frac{1}{x-1} \) approaches 0 and \( -2x \) becomes very large negatively. Since the term \( -2x \) dominates, there is no horizontal asymptote as the output goes to negative infinity.

Key Concepts

Vertical Asymptotes

When dealing with functions that involve rational expressions, such as this one, vertical asymptotes occur where the function is undefined. Specifically, vertical asymptotes are found wherever the denominator of any fraction in the function equals zero. This leads to division by zero, which makes the function undefined at that point.

In our original function, given by:

• \(f(x) = \frac{1}{x-1} - 2x\)

we focus on the fractional part \(\frac{1}{x-1}\). The denominator here is \(x-1\). To find the vertical asymptote, set it to zero:

• \(x - 1 = 0\)
• Solve for \(x\): \(x = 1\).

Therefore, at \(x = 1\), the function has a vertical asymptote. This means as \(x\) approaches 1 from either left or right, the value of \(f(x)\) will go to infinity or negative infinity.

Vertical asymptotes tell us important information about the behavior of a graph at specific x-values. They act like invisible lines indicating values that the graph will get infinitely close to but never actually cross. As you study functions, identifying these lines helps in understanding where a function is impossible to define, thus better predicting the graph's behavior.

Horizontal Asymptotes

Horizontal asymptotes are primarily concerned with the behavior of a function as \(x\) approaches very large positive or negative numbers (infinity or negative infinity). They suggest a trend where the function levels off to a specific value as \(x\) grows toward infinity.

In this example, our function is:

• \(f(x) = \frac{1}{x-1} - 2x\)

To determine if a horizontal asymptote exists, analyze the dominant terms as \(x\) approaches infinity. The term \(\frac{1}{x-1}\) becomes very small because the division with large \(x\) values makes it approach 0. Meanwhile, the term \(-2x\) increases rapidly in size in the negative direction.

Since the term \(-2x\) dominates the behavior of the function at infinity, \(-2x\) dictates that \(f(x)\) goes to negative infinity as \(x\) goes to infinity. Hence, there isn't a horizontal asymptote; the function doesn’t level off at a particular horizontal line. This situation illustrates that not all functions will have a horizontal asymptote and understanding the behavior of the highest order terms gives insight into this aspect of a function.

Functions

A function is a special relation between input and output, where each input is linked to exactly one output. Functions are foundational in mathematics and are used to represent real-world phenomena.

Functions in algebra are written as \(f(x)\), which denotes a rule applied to \(x\) values to produce corresponding \(f(x)\) values. In this exercise, the function is given by:

• \(f(x) = \frac{1}{x-1} - 2x\)

Here, it includes a rational term \(\frac{1}{x-1}\) and a linear term \(-2x\). Recognizing different components of functions is crucial:

• **Rational Functions** include fractions and are key in finding asymptotic behavior. They must be managed carefully due to division.
• **Linear Functions** have a constant rate of change, showcased simply by terms like \(2x\).

Functions give us powerful tools not just to calculate numbers but to model and solve complex real-life situations. Whether predicting growth trends, managing resources, or understanding physical laws, mastering functions is fundamental. Understanding how they behave around their asymptotes and their general behavior helps you predict how graphs unfold.