Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. $$ g(x)=\frac{14}{2 x^{2}+7} $$

Step-by-Step Solution

Answer

No vertical asymptotes; horizontal asymptote is \( y = 0 \).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the expression is undefined. For the function \( g(x) = \frac{14}{2x^2 + 7} \), we need to solve \( 2x^2 + 7 = 0 \). However, since \( 2x^2 + 7 \) is always positive for all real \( x \), as \( 2x^2 \) is always non-negative and adds to 7, there are no real values of \( x \) making this zero. Thus, there are no vertical asymptotes.

Step 2: Identify Horizontal Asymptotes

Horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator. Here, the degree of the numerator (0) is less than the degree of the denominator (2). When the numerator's degree is less than the denominator's, the horizontal asymptote is \( y = 0 \). Thus, the horizontal asymptote is \( y = 0 \).

Step 3: Sketching the Graph

To sketch the graph, note the function approaches \( y = 0 \) as \( x \) becomes very large or very small (positive or negative infinity), due to the horizontal asymptote. Since there are no vertical asymptotes, the function is smooth and continuous across all x-values. The graph will be symmetric around the y-axis because \( g(x) \) is an even function. Consider evaluating at some key points to further illustrate the curve's behavior.

Key Concepts

Vertical Asymptotes

In calculus and algebra, vertical asymptotes occur where a rational function becomes undefined. This usually happens when the denominator is set to zero. For our function, \( g(x) = \frac{14}{2x^2 + 7} \), this would require solving \( 2x^2 + 7 = 0 \) to find any real values of \( x \) that make it zero. However, this equation has no real solutions because \( 2x^2 + 7 \) is always positive for all real \( x \). Therefore, this function does not have any vertical asymptotes. This means that the graph of the function will not exhibit any infinite spikes or breaks, ensuring a smooth curve across its domain.

Horizontal Asymptotes

A horizontal asymptote refers to a horizontal line that a graph approaches as \( x \) heads towards infinity or negative infinity. To find horizontal asymptotes of a function, we compare the degrees of the polynomial in the numerator and the denominator. For \( g(x) = \frac{14}{2x^2 + 7} \):

• The degree of the numerator is 0, since it is a constant \( (14) \).
• The degree of the denominator is 2, given by \( 2x^2 \).

When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \). Therefore, the graph of \( g(x) \) approaches the x-axis, but never touches or crosses it, as \( x \) moves towards positive or negative infinity.

Function Graphing

Graphing the function \( g(x) = \frac{14}{2x^2 + 7} \) involves identifying its behavior as it relates to asymptotes and any key characteristics, like symmetry. Since the horizontal asymptote is \( y = 0 \), the graph gets closer to this line but never touches it as \( x \) becomes very large in the positive or negative direction.
Additionally, with no vertical asymptotes, the function is defined for all real numbers, meaning the graph will have no breaks or jumps.
When sketching, plotting some points can further showcase the nature of the curve. Consider evaluating the function at different values of \( x \) to get a better idea of the graph's shape.

Even Functions

Even functions are symmetrical about the y-axis. Mathematically, a function \( f(x) \) is even if \( f(x) = f(-x) \) for all \( x \) in its domain. For \( g(x) = \frac{14}{2x^2 + 7} \), substituting \( -x \) for \( x \) does not change the value of the function:

• \( g(-x) = \frac{14}{2(-x)^2 + 7} = \frac{14}{2x^2 + 7} = g(x) \)

As a result, \( g(x) \) is an even function, and its graph will be mirror-symmetrical about the y-axis. This symmetry simplifies graphing since knowing the graph's shape on the positive side of the x-axis helps predict its shape on the opposite side.