Problem 24

Question

Find the horizontal asymptote of the graph of the function. Then sketch the graph of the function. $$ f(x)=-1 /(x+3) $$

Step-by-Step Solution

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Answer

The horizontal asymptote is $ y = 0 $.

1Step 1: Identify the Function Type

The given function is a rational function, which is of the form $ f(x) = \frac{p(x)}{q(x)} $, where $ p(x) $ and $ q(x) $ are polynomials. Here, $ p(x) = -1 $ and $ q(x) = x + 3 $. A rational function can have horizontal, vertical, or oblique asymptotes.

2Step 2: Find the Horizontal Asymptote

For rational functions where the degree of the numerator $ p(x) $ is less than the degree of the denominator $ q(x) $, the horizontal asymptote is $ y = 0 $. In this case, since the degree of $ -1 $ (which is 0) is less than the degree of $ x+3 $ (which is 1), the horizontal asymptote is $ y = 0 $.

3Step 3: Sketch the Graph

To sketch the graph, recognize that $ f(x) = -\frac{1}{x+3} $ is a hyperbola. The function has a vertical asymptote at $ x = -3 $. As $ x \to -\infty $ or $ x \to \infty $, the function approaches the horizontal asymptote $ y = 0 $. Plot points and ensure the graph reflects the behavior of moving towards these asymptotes. The graph will decrease from left to the right across the vertical asymptote, above the horizontal asymptote for large negative $ x $, and below it for large positive $ x $.

Key Concepts

Rational FunctionsAsymptotesGraph Sketching

Rational Functions

Rational functions are quite common in mathematics and they are expressed as the ratio of two polynomials. In simpler terms, you can think of them as fractions where the numerator and the denominator are both polynomials. This gives them a distinct characteristic of having varied asymptotic behaviors, depending on the degrees of these polynomials. A rational function generally takes the form $ f(x) = \frac{p(x)}{q(x)} $, which means:

$ p(x) $ is a polynomial in the numerator.
$ q(x) $ is a polynomial in the denominator.

In our specific example, the function is $ f(x) = -\frac{1}{x+3} $ where the numerator $ p(x) = -1 $ and the denominator $ q(x) = x+3 $. Understanding the ratio format helps in identifying asymptotic behavior and guides the graph sketching process.

Asymptotes

When it comes to rational functions, asymptotes are crucial lines that the graph approaches but never actually touches. There are three main types of asymptotes to consider:

Horizontal Asymptotes: These indicate the behavior of the curve as it stretches infinitely to the left or right. For the function $ f(x) = -\frac{1}{x+3} $, the horizontal asymptote is determined by checking the degrees of the numerator and denominator. Because the degree of $ -1 $ (numerator) is less than the degree of $ x+3 $ (denominator), the horizontal asymptote is at $ y = 0 $.
Vertical Asymptotes: These are found where the denominator equals zero, indicating points where the function is undefined. For $ f(x) = -\frac{1}{x+3} $, the vertical asymptote is at $ x = -3 $ because dividing by zero is undefined.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. However, these are not applicable in our case.

Recognizing and understanding these asymptotes can significantly aid in sketching the graph accurately.

Graph Sketching

Sketching the graph of a rational function brings together all the information you have found about the function's behavior. Knowing the asymptotes, you can better see how the function behaves near these invisible boundaries. To sketch the function $ f(x) = -\frac{1}{x+3} $:

Start by plotting the vertical asymptote $ x = -3 $. The graph cannot cross this line and will bend towards it.
Draw the horizontal asymptote $ y = 0 $. The function will approach this line as $ x \to infinite $ positive or negative values.
Select strategic points around the vertical asymptote and evaluate the function to see on which side the graph resides. For $ x < -3 $, the graph will be above the horizontal asymptote, whereas for $ x > -3 $, it will fall below it.

By reflecting these points and behaviors on the graph, you create a "hyperbolic" shape moving from left to right, maintaining respect to the vertical and horizontal asymptotes. Understanding each segment of the curve is key to mastering graph sketching and embodying the movement and limits defined by the asymptotes.

Problem 23

Problem 24

Other exercises in this chapter

Problem 23

Find all extreme values (if any) of the given function on the given interval. Determine at which numbers in the interval these values occur. $$ k(z)=\sqrt{1+z^{

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Problem 23

Determine all functions $f$ satisfying the given conditions. $$ f^{\prime \prime}(x)=0, f^{\prime}(0)=-1, f(0)=2 $$

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Problem 24

Find all inflection points (if any) of the graph of the function. Then sketch the graph of the function. $$ f(x)=x^{3}-\frac{3}{2} x^{2}-6 x $$

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Problem 24

Use the First Derivative Test to determine the relative extreme values (if any) of the function. $$ f(x)=\ln x-\ln \left(x^{2}+1\right) $$

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