Problem 43 - 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. \(f(x)=\frac{3}{x+1}\)

Step-by-Step Solution

Answer

Vertical asymptote at \( x = -1 \); horizontal asymptote at \( y = 0 \).

Step 1: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function is equal to zero and the numerator is non-zero. For the given function \( f(x) = \frac{3}{x+1} \), set the denominator equal to zero: \( x + 1 = 0 \). Solving this gives \( x = -1 \). Thus, there is a vertical asymptote at \( x = -1 \).

Step 2: Determine Horizontal Asymptotes

Horizontal asymptotes are determined by the behavior of the function as \( x \) approaches infinity or negative infinity. For the function \( f(x) = \frac{n(x)}{d(x)} = \frac{3}{x+1} \), where the degree of the numerator \( n(x) = 0 \) and the degree of the denominator \( d(x) = 1 \), if \( ext{degree of } n(x) < ext{degree of } d(x) \), then the horizontal asymptote is \( y = 0 \). Therefore, there is a horizontal asymptote at \( y = 0 \).

Step 3: Sketch the Graph

With a vertical asymptote at \( x = -1 \) and a horizontal asymptote at \( y = 0 \), we can sketch the graph. As \( x \) approaches \( -1 \), the function grows to infinity or negative infinity, depending on the direction from which \( x \) is approaching. For \( x > -1 \), the graph is above the x-axis and for \( x < -1 \), it is below the x-axis, approaching \( y = 0 \) as \( x \) moves towards positive or negative infinity.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are found where the denominator of a rational function equals zero, provided the numerator isn’t zero at the same point. In the function \( f(x) = \frac{3}{x+1} \), a vertical asymptote occurs when \( x+1 = 0 \). Solving this yields \( x = -1 \). This means the line \( x = -1 \) is where the graph cannot exist, as the function becomes undefined. When the graph is close to \( x = -1 \), the y-values of the function shoot up to infinity or down to negative infinity. This behavior results in the graph approaching but never touching the line \( x = -1 \).
Vertical asymptotes serve as invisible lines that separate the graph into distinct sections, each influenced by these boundaries.

Horizontal Asymptotes

Horizontal asymptotes describe how a function behaves as \( x \) approaches very large or very small values. For \( f(x) = \frac{3}{x+1} \), we examine the degrees of the polynomial in the numerator and the polynomial in the denominator. Here, the numerator is a constant (degree 0), and the denominator is of degree 1.
According to asymptote rules, if the degree of the numerator is less than that of the denominator, the horizontal asymptote is \( y = 0 \). This means as \( x \) moves far left or far right, the graph flattens closer and closer to the x-axis. However, it will never actually touch or cross \( y = 0 \).

• This behavior indicates that no matter the value of \( x \), the function’s output does not grow infinitely but stabilizes towards a constant line.

Graph Sketching

Graph sketching involves understanding the asymptotic behavior and plotting the curve. To chart \( f(x) = \frac{3}{x+1} \), start with the asymptotes: a vertical line at \( x = -1 \) and a horizontal line at \( y = 0 \). These guide the graph's path.
Now, plot a few points around \( x = -1 \) to visualize how the graph tracks these lines. For instance:

• For \( x = 0 \), \( f(x) = 3 \) indicating the point (0, 3).
• For \( x = -2 \), \( f(x) = -3 \) indicating the point (-2, -3).

Notice how negative \( x \) values result in negative \( f(x) \) values, showing how the graph approaches from below before coming close to the asymptote \( y = 0 \) as \( x \) heads to negative infinity.
Graph sketching focuses not just on plotting points, but on appreciating how the function behaves near and at its limits, guided by the asymptotes. This helps draw a more precise picture of the graph.