Question

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. $$ f(x)=\frac{3}{x+1} $$

Short answer

Vertical asymptote: \(x = -1\); Horizontal asymptote: \(y = 0\).

Step-by-step solution

Identifying Vertical Asymptotes

To find the vertical asymptotes of the function \(f(x) = \frac{3}{x+1}\), look for values of \(x\) that make the denominator zero. Here, the denominator is \(x + 1\). Setting \(x + 1 = 0\) gives \(x = -1\). Hence, there is a vertical asymptote at \(x = -1\).

Identifying Horizontal Asymptotes

To find the horizontal asymptote, compare the degrees of the polynomials in the numerator and denominator. In \(f(x) = \frac{3}{x+1}\), the degree of the numerator is 0 and the degree of the denominator is 1. Since the degree of the denominator is greater, the horizontal asymptote is at \(y = 0\).

Graph Sketching

To sketch the graph, plot the vertical asymptote at \(x = -1\) and the horizontal asymptote at \(y = 0\). Since \(f(x) = \frac{3}{x+1}\) is a hyperbola, graph it showing a branch in each quadrant near the asymptotes: going up to infinity in Quadrant II and down to negative infinity in Quadrant IV as \(x\) approaches \(x = -1\). The function approaches \(y = 0\) horizontally as \(x\) goes to positive or negative infinity.

Key concepts

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches or crosses. They act as boundaries for the graph of a function. There are mainly two types of asymptotes to consider in rational functions: vertical and horizontal.

A vertical asymptote forms at the values of \(x\) where the denominator of a rational function equals zero, creating a division by zero. For the given function \(f(x) = \frac{3}{x+1}\), setting the denominator "\(x + 1\)" to zero gives \(x = -1\).
This means there is a vertical asymptote at \(x = -1\). Vertical asymptotes indicate that as the graph approaches \(x = -1\), the function's value becomes very large, going towards positive or negative infinity.

Horizontal asymptotes are determined by the degrees of the numerator and denominator of a rational function. For \(f(x) = \frac{3}{x+1}\), the numerator's degree is 0, while the denominator's degree is 1. Because the denominator's degree is higher, the horizontal asymptote is \(y = 0\).
This suggests that as \(x\) goes to positive or negative infinity, the function's value creeps closer to, but never quite reaches, zero.

Rational Functions

Rational functions are the ratios of two polynomials. They can be written in the form \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\). These functions are defined by their constraints and behavior around their asymptotes.

In the example of \(f(x) = \frac{3}{x+1}\), the constant numerator (3) determines the function's overall behavior. Because there are no \(x\) terms in the numerator, the function decreases as a hyperbola. This particular rational function is defined everywhere except where the denominator is zero, which is at \(x = -1\).
This is where its vertical asymptote appears.

Rational functions often have real-life applications in various fields like engineering and physics, especially in scenarios where division or proportions are involved. They are key for understanding the concept of limits and continuity in calculus.

Graph Sketching

Graph sketching involves drawing a curve based on observable features from its function equation, asymptotes, and overall behavior. It's a crucial skill to visualize the function's behavior and trends.

For the function \(f(x) = \frac{3}{x+1}\), start by drawing its vertical asymptote at \(x = -1\) and its horizontal asymptote at \(y = 0\). The asymptotes help to divide the graph into segments and guide the sketch's form.

In this case, since the function resembles a hyperbola, you expect it to have properties like symmetry and infinite tails. In the sketch for this function:

• In Quadrant II (top left corner of the Cartesian plane), the graph approaches \(x = -1\) going up toward positive infinity.
• In Quadrant IV (bottom right corner), it approaches \(x = -1\) going down toward negative infinity.
• As \(x\) moves away from the vertical asymptote, the graph tends closer to the horizontal asymptote \(y = 0\).

Graph sketching based on asymptotes provides a powerful way to understand and communicate a function's general shape and extreme behavior without needing to plot numerous points.