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 8

Chapter 4, Problem 8

Use the graph to solve each equation or inequality. Use interval notation where appropriate. 2(x-2) / {(x-1)(x-3)} < 0

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Identify the critical points of the rational expression $\frac{2(X-2)}{(X-1)(X-3)}$. These points are where the numerator or denominator equals zero: $X=2$ (numerator zero), $X=1$ and $X=3$ (denominator zero, vertical asymptotes).

Divide the number line into intervals based on these critical points: $(-\infty, 1)$, $(1, 2)$, $(2, 3)$, and $(3, \infty)$.

Determine the sign of the expression $\frac{2(X-2)}{(X-1)(X-3)}$ on each interval by choosing a test point from each interval and substituting it into the expression.

Use the graph to verify the sign of the function on each interval. The graph shows the function's behavior around the vertical asymptotes at $X=1$ and $X=3$ and the zero at $X=2$.

Select the intervals where the expression is less than zero (negative) and write the solution in interval notation, excluding points where the denominator is zero (since the function is undefined there).

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Key Concepts

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

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the rational expression is positive, negative, or zero by analyzing the signs of numerator and denominator.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, causing the rational expression to be zero or undefined. These points divide the number line into intervals, and testing each interval helps determine where the inequality holds true.

Graph Interpretation of Rational Functions

Graphs of rational functions show vertical asymptotes at values making the denominator zero and zeros where the numerator is zero. By examining the graph, one can identify intervals where the function is above or below the x-axis, aiding in solving inequalities.

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Related Practice

Textbook Question

Use the graphs of the rational functions in choices A–D to answer each question.

There may be more than one correct choice. Which choices have domain (-∞, 3)U(3, ∞)?

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Textbook Question

Solve each problem. During the course of a year, the number of volunteers available to run a food bank each month is modeled by $V (x)$ , where $V (x) = 2 x^{2} - 32 x + 150$ between the months of January and August. Here x is time in months, with x=1 representing January. From August to December, $V (x)$ is modeled by $V (x) = 31 x - 226$ . Find the number of volunteers in each of the following months. Sketch a graph of $y = V (x)$ for January through December. In what month are the fewest volunteers available?

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Textbook Question

Determine whether each statement is true or false. If false, explain why. The product of a complex number and its conjugate is always a real number.

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Textbook Question

Provide a short answer to each question. Is ƒ(x)=1/x an even or an odd function? What symmetry does its graph exhibit?

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Textbook Question

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=2x⁴

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Textbook Question

December

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