Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 31

Chapter 4, Problem 31

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. ƒ(x)=1/(x+4)

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Identify the given rational function: $f(x) = \frac{1}{x+4}$.

Recognize that the function is a transformation of the parent function $f(x) = \frac{1}{x}$, shifted horizontally.

Determine the vertical asymptote by setting the denominator equal to zero: $x + 4 = 0$, which gives $x = -4$.

Note that the horizontal asymptote of the function remains $y = 0$ because the degree of the numerator is less than the degree of the denominator.

Match the function to the description that mentions a vertical asymptote at $x = -4$ and a horizontal asymptote at $y = 0$, indicating a horizontal shift of the parent function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the form and behavior of rational functions helps in identifying their key features such as asymptotes and domain restrictions.

Domain of Rational Functions

The domain of a rational function includes all real numbers except where the denominator equals zero. For f(x) = 1/(x+4), the domain excludes x = -4, since division by zero is undefined, which is critical for matching the function to its description.

Vertical and Horizontal Asymptotes

Vertical asymptotes occur where the denominator is zero, indicating values the function cannot take. Horizontal asymptotes describe the end behavior of the function as x approaches infinity or negative infinity. For f(x) = 1/(x+4), x = -4 is a vertical asymptote, and y = 0 is a horizontal asymptote.

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Determining Horizontal Asymptotes

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Textbook Question

Distance to the Horizon The distance that a person can see to the horizon on a clear day from a point above the surface of Earth varies directly as the square root of the height at that point. If a person 144 m above the surface of Earth can see 18 km to the horizon, how far can a person see to the horizon from a point 64 m above the surface?

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Textbook Question

Factor $ƒ (x)$ into linear factors given that k is a zero. $ƒ (x) = x^{4} + 2 x^{3} - 7 x^{2} - 20 x - 12; k = - 2$ (multiplicity $2$ )

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Textbook Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 Find the zero in part (b) to three decimal places.

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Textbook Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 between 2 and 3

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Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. -(x - 3)(x - 4)² (x - 5) > 0

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Textbook Question

Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=3x^3-8x^2+x+2 between -1 and 0

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