Problem 89 - Algebra and Trigonometry

Question

Constructing a Rational Function from Its Asymptotes Give an example of a rational function that has vertical asymptote \(x=3\) . Now give an example of one that has vertical asymptote \(x=3\) and horizontal asymptote \(y=2\) Now give an example of a rational function with vertical asymptotes \(x=1\) and \(x=-1,\) horizontal asymptote \(y=0,\) and \(x\) -intercept \(4 .\)

Step-by-Step Solution

Answer

Examples: 1) \(f(x) = \frac{1}{x-3}\), 2) \(f(x) = \frac{2x}{x-3}\), 3) \(f(x) = \frac{(x-4)}{(x-1)(x+1)}\).

Step 1: Understanding Vertical Asymptote

A vertical asymptote at \(x = 3\) means the function has a factor \((x - 3)\) in the denominator. An example of a rational function with this asymptote is \(f(x) = \frac{1}{x - 3}\).

Step 2: Adding Horizontal Asymptote

A horizontal asymptote at \(y = 2\) indicates that as \(x\) approaches infinity, the value of the function should approach 2. This can be achieved by having the leading coefficients of the numerator and denominator equal when they have the same degree. An example is \(f(x) = \frac{2x}{x - 3}\).

Step 3: Multiple Vertical Asymptotes

For vertical asymptotes at \(x = 1\) and \(x = -1\), the factors \((x - 1)\) and \((x + 1)\) must be in the denominator. Therefore, an example function is \(f(x) = \frac{x}{(x - 1)(x + 1)}\).

Step 4: Aligning the Horizontal Asymptote

The given horizontal asymptote \(y = 0\) implies the degree of the numerator must be less than the degree of the denominator, as \(x\) approaches infinity. An example consistent with this is \(f(x) = \frac{x}{(x - 1)(x + 1)}\).

Step 5: Incorporating x-intercept

For an \(x\)-intercept at \(x = 4\), the numerator must be \(0\) when \(x = 4\). Thus, the numerator should include the factor \((x - 4)\). An example of a function with the specified criteria is \(f(x) = \frac{(x - 4)}{(x - 1)(x + 1)}\).

Key Concepts

Asymptotes

Asymptotes are invisible lines that guide the behavior of a graph, but which the graph never actually crosses. They come in different forms, most commonly vertical and horizontal. Understanding asymptotes is pivotal when dealing with rational functions.
As you explore the graph of a rational function, these invisible boundaries help determine the direction and shape of the curve.

• Vertical Asymptotes: The graph will approach but never actually touch this line, usually because the denominator of the function approaches zero at this point, causing the function to head towards infinity.
• Horizontal Asymptotes: This is where the graph levels off as x heads towards infinity or negative infinity, indicating a finite y-value the function is approaching.

Knowing how asymptotes influence the graph helps immensely in sketching and understanding these functions.

Vertical Asymptote

A vertical asymptote occurs at a certain value of x, making the function 'explode' towards infinity, meaning the function is undefined at this point. This happens when the denominator of a rational function equals zero.
Let's say a rational function has a vertical asymptote at \(x = 3\). This implies there's a factor \((x - 3)\) in the denominator and the function behaves acutely around this line.For a simple example, consider the function:

• \(f(x) = \frac{1}{x - 3}\)

This function will have a pronounced vertical asymptote at \(x = 3\), as the denominator is zero there, causing the graph to reach extreme values near this line. Understanding vertical asymptotes helps you predict where the graph does something drastically different.

Horizontal Asymptote

Horizontal asymptotes inform you about the end behavior of a function as x becomes very large or very small. This kind of asymptote tells you the y-value that the graph approaches but never touches as x tends towards infinity or negative infinity.
Consider the example of a rational function with a horizontal asymptote at \(y = 2\). This can happen when the degrees of the polynomial in the numerator and the denominator are the same, and the ratio of the leading coefficients equals 2.For instance:

• \(f(x) = \frac{2x}{x - 3}\)

As \(x\) approaches infinity, this function approaches \(y = 2\), creating a horizontal asymptote.
Horizontal asymptotes give you insights into what happens as the graph stretches endlessly left or right, providing a picture of its broad behavior.

x-intercept

The x-intercept of a rational function is the point where the graph crosses the x-axis, which means the output of the function is zero at this x-value. To find this, set the numerator of the rational function equal to zero because that's the point where the function equals zero.
Suppose you're given a function and you need an x-intercept at \(x = 4\). The numerator should then include the factor \((x - 4)\) so that replacing x with 4 makes the numerator zero.Consider an example:

• \(f(x) = \frac{(x - 4)}{(x - 1)(x + 1)}\)

Here, the x-intercept is at \(x = 4\) since the numerator is zero when \(x = 4\).Finding x-intercepts helps you determine key points where the graph touches or crosses the axis, providing important context for graphing and function analysis.