Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. Do not use a calculator. \(\mathbf{I}\) \(f(x)=\frac{1}{x+12}\) \(\mathbf{II}\) A. The \(x\) -intercept is \((-3,0)\) B. The \(y\) -intercept is \((0,5)\) C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a hole in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote. H. The \(y\) -axis is its vertical asymptote.

Step-by-Step Solution

Answer

The horizontal asymptote is the x-axis, option G applies to this function.

Step 1: Identify Form of the Rational Function

The given rational function is \( f(x) = \frac{1}{x+12} \). This is a standard form \( \frac{1}{x-a} \) of a rational function, indicating potential characteristics like vertical asymptotes and horizontal asymptotes.

Step 2: Determine Vertical Asymptote

Vertical asymptotes occur when the denominator is zero (and the numerator is not zero at that same point). For \( f(x) = \frac{1}{x+12} \), set the denominator equal to zero: \( x+12 = 0 \). Thus, the vertical asymptote is \( x = -12 \).

Step 3: Determine Horizontal Asymptote

For a rational function \( \frac{p(x)}{q(x)} \), if the degree of \( p(x) \) is less than the degree of \( q(x) \), the horizontal asymptote is \( y = 0 \). Here, \( f(x) = \frac{1}{x+12} \) has a horizontal asymptote at \( y = 0 \) since the numerator degree (0) is less than the denominator degree (1). This corresponds to the \( x \)-axis being the horizontal asymptote.

Key Concepts

Vertical Asymptotes

In rational functions, vertical asymptotes are lines where the function approaches infinity. This happens when the denominator of the function becomes zero, and the numerator isn’t zero at that exact point. For instance, in the function \( f(x) = \frac{1}{x+12} \), we find vertical asymptotes by setting the denominator \( x+12 \) equal to zero, which solves to \( x = -12 \).

This means as \( x \) gets very close to \(-12\), the function's value spikes up to positive or negative infinity. It’s important to note that vertical asymptotes are not part of the graph. These are merely "guidelines" that show where the function cannot exist due to division by zero. Keep in mind that vertical asymptotes differ from holes, a topic we won't delve into here, which involve canceling factors in the numerator and denominator.

Horizontal Asymptotes

Horizontal asymptotes are straight lines that a rational function approaches as the input \( x \) goes to infinity or negative infinity. They indicate the end behavior of a graph. For rational functions like \( f(x) = \frac{1}{x+12} \), the horizontal asymptote depends on the degree of the polynomials in the numerator and denominator.

Here's a simple rule to remember:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, the asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator, respectively.
• If the numerator’s degree is greater, there generally isn't a horizontal asymptote.

Applying this to \( f(x) = \frac{1}{x+12} \), since the numerator's degree (0) is less than the denominator's degree (1), we find that the horizontal asymptote is \( y = 0 \).

X-intercepts

The \( x \)-intercept of a function is the point where the graph crosses the \( x \)-axis. It can be found by setting the function equal to zero and solving for \( x \). In rational functions, \( x \)-intercepts occur when the numerator is zero, provided that it does not make the denominator zero at the same time.

For the function \( f(x) = \frac{1}{x+12} \), notice the numerator is a constant, 1, which never equals zero. This means \( f(x) \) has no \( x \)-intercepts. When there are no \( x \)-intercepts, the graph never crosses the \( x \)-axis. This is often the case for functions with non-zero constant numerators. Understanding how to find \( x \)-intercepts helps us anticipate these crossings, or lack thereof, in more complex functions.