Problem 57 - Precalculus: Functions and Graphs

Question

\( Graph f,\) and find equations of the vertical asymptotes. $$f(x)=\frac{(x-1)^{2}}{(x-0.999)^{2}}$$

Step-by-Step Solution

Answer

The vertical asymptote is \(x = 0.999\).

Step 1: Analyze the Denominator

The function \( f(x) = \frac{(x-1)^{2}}{(x-0.999)^{2}} \) is undefined when the denominator is zero. Let's set the denominator equal to zero to find the points of discontinuity: \((x-0.999)^2 = 0\). Solving this equation gives \(x-0.999 = 0\). Thus, \(x = 0.999\).

Step 2: Determine Points of Discontinuity

Since the denominator becomes zero at \(x=0.999\), we need to verify if it is indeed a vertical asymptote. A vertical asymptote is a point where the function approaches infinity. Here, \(x=0.999\) makes the function undefined and is a vertical asymptote because the denominator is a higher power than the numerator, causing the function to explode to infinity.

Step 3: Verify the Vertical Asymptote

Now, check around \(x=0.999\). As \(x\) approaches \(0.999\) from the left or right, the function tends towards infinity: \(f(x) = \frac{\text{positive number}}{\text{very close to zero}}\). This confirms \(x = 0.999\) is a vertical asymptote.

Step 4: Conclusion

The equation of the vertical asymptote is \(x = 0.999\). There are no horizontal asymptotes because the power in the numerator is lower than in the denominator.

Key Concepts

Rational Functions

A rational function is a fraction where both the numerator and the denominator are polynomials. For example, the function \( f(x) = \frac{(x-1)^{2}}{(x-0.999)^{2}} \) given in the exercise is a rational function because both the numerator

• \( (x-1)^2 \)
• the denominator \( (x-0.999)^2 \)

are polynomial expressions. Understanding rational functions is crucial in precalculus, as they form the basis for analyzing more complex equations. The key aspect of rational functions is determining where they are defined, which involves analyzing when the denominator equals zero, thus making the function undefined at certain points.

Discontinuities

Discontinuities in a function occur when there are breaks, jumps, or holes in its graph. In the context of rational functions, discontinuities are primarily found where the denominator is zero. For the function \( f(x) = \frac{(x-1)^{2}}{(x-0.999)^{2}} \), the main discontinuity appears at \( x = 0.999 \).

• This is because setting the denominator \((x-0.999)^2 = 0\) reveals when the function becomes undefined.

To understand how the graph behaves at this point, it is necessary to consider both sides of the discontinuity: as \( x \) approaches \( 0.999 \) from the left and right. Since the function approaches infinity from both sides, this results in a vertical asymptote.

Asymptotic Behavior

Asymptotic behavior describes how a function acts as it approaches certain points or infinity. Vertical asymptotes occur where a function grows unbounded as \( x \) approaches a particular value. For the function \( f(x) = \frac{(x-1)^{2}}{(x-0.999)^{2}} \), the vertical asymptote is at \( x = 0.999 \). Asymptotic behavior helps us understand and visualize what happens near these points:

• When the denominator is a higher power than the numerator, the function typically tends towards infinity, causing a vertical asymptote.

These asymptotes are important in sketching the graph of the function and in understanding how the function behaves around discontinuities.

Precalculus

Precalculus is the mathematical groundwork that prepares students for calculus. It covers a variety of topics including functions, algebra, and trigonometry. In precalculus, students learn to analyze and graph rational functions, such as \( f(x) = \frac{(x-1)^{2}}{(x-0.999)^{2}} \), understanding their characteristics and discontinuities.
One fundamental goal in precalculus is to master the concept of asymptotes and discontinuities. Knowing how to find vertical asymptotes, like \( x = 0.999 \), allows students to graph these functions accurately. Mastering these concepts provides a smooth transition to the more complex ideas in calculus, such as limits and integrals. Precalculus enables students to see the bigger picture in mathematical analysis and function behavior.