Problem 17 - Algebra 2

Question

Graph each rational function. $$ f(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The graph features a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\), forming two branches reflecting over the origin.

Step 1: Determine the Domain

The function is given as \(f(x) = \frac{1}{x}\). Rational functions are undefined where the denominator is zero. Therefore, the domain of this function is all real numbers except where \(x = 0\). Thus, the domain is \(x eq 0\).

Step 2: Identify the Vertical Asymptote

A vertical asymptote occurs where the function is undefined. In this case, the denominator is zero when \(x = 0\), so there is a vertical asymptote at \(x = 0\).

Step 3: Identify the Horizontal Asymptote

For large values of \(x\), the behavior of \(f(x) = \frac{1}{x}\) can be analyzed. As \(x\) approaches infinity or negative infinity, \(f(x)\) approaches zero. Therefore, the horizontal asymptote is \(y = 0\).

Step 4: Determine Key Points

Choose a few easy points to calculate, such as \(x = 1, -1, 2, -2\). Calculating these gives: \(f(1) = 1\), \(f(-1) = -1\), \(f(2) = 0.5\), \(f(-2) = -0.5\). These points help sketch the graph.

Step 5: Sketch the Graph

To sketch the graph, plot the key points: (1, 1), (-1, -1), (2, 0.5), (-2, -0.5). Draw a curve approaching the vertical asymptote at \(x = 0\) and the horizontal asymptote at \(y = 0\). The function is in two branches, one in the first quadrant and another in the third quadrant, both reflecting over the origin.

Key Concepts

Vertical Asymptotes

Vertical asymptotes of a rational function occur where the denominator of the function is equal to zero and the numerator is not zero at the same point. This results in a value of the function approaching infinity or negative infinity. For the given function, which is \( f(x) = \frac{1}{x} \), the denominator, \( x \), equals zero when \( x = 0 \). Consequently, at this point, the function is undefined and the graph will shoot up to or drop down towards infinity, indicating the existence of a vertical asymptote at \( x = 0 \).

Here's how you can identify vertical asymptotes in general:

• Set the denominator equal to zero and solve for \( x \).
• Check that the numerator is not zero at these solutions, since it wouldn’t be an asymptote if both were zero (a hole would appear instead).

As you graph \( f(x) = \frac{1}{x} \), you’ll note the disruption at \( x = 0 \), which helps visualize how the function behaves near vertical asymptotes.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as \( x \) approaches very large positive or negative values. Unlike vertical asymptotes, horizontal ones indicate the end behavior of the function, rather than places where it becomes undefined. For \( f(x) = \frac{1}{x} \), as \( x \) heads towards infinity or negative infinity, the output \( f(x) \) gets closer and closer to zero. Thus, the horizontal asymptote for this function is \( y = 0 \).

Consider these tips when determining horizontal asymptotes:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator equals the degree of the denominator, divide the leading coefficients to find the asymptote.
• If the numerator's degree exceeds the denominator's degree, a horizontal asymptote does not exist.

Understanding horizontal asymptotes is crucial when examining how functions behave at extremes.

Domain of a Function

The domain of a rational function specifies all the possible input values \( x \) can take without making the function undefined. This usually involves identifying the values that make the denominator zero, since these will not be in the domain. For the function \( f(x) = \frac{1}{x} \), the denominator becomes zero at \( x = 0 \). This point must be excluded from the domain, meaning the domain is all real numbers except \( x = 0 \).

To determine the domain of any rational function:

• Set the denominator equal to zero and solve for \( x \).
• Exclude these solutions from the set of all real numbers, using interval notation or inequality notation to define the domain.

For \( f(x) = \frac{1}{x} \), the domain is often written as \( x eq 0 \) or \( (-\infty, 0) \cup (0, \infty) \), providing a clear picture of where the function can accept real number inputs.