Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation.
(a) -(x + 1)(x + 2) ≥ 0
(b) -(x + 1)(x + 2) > 0
(c) -(x + 1)(x + 2) ≤ 0
(d) -(x + 1)(x + 2) < 0
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 13
Use the graphs of the rational functions in choices A–D to answer each question.
There may be more than one correct choice. If ƒ represents the function, only one choice has a single solution to the equation ƒ(x)=3. Which one is it?
Verified step by step guidance1
Step 1: Understand the problem. We need to find which graph has exactly one solution to the equation \(f(x) = 3\). This means we are looking for the graph where the horizontal line \(y = 3\) intersects the function's graph exactly once.
Step 2: Analyze each graph by visualizing or drawing the horizontal line \(y = 3\) and count the number of intersection points with the graph of \(f(x)\).
Step 3: For graph A, observe that the curve crosses the line \(y = 3\) exactly once. This suggests one solution for \(f(x) = 3\).
Step 4: For graph B, notice that the curve crosses the line \(y = 3\) more than once, indicating multiple solutions to \(f(x) = 3\).
Step 5: For graphs C and D, similarly check the number of intersections with \(y = 3\). Both graphs show two or more intersections, meaning multiple solutions. Therefore, only graph A has a single solution to \(f(x) = 3\).
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions and Their Graphs
A rational function is a ratio of two polynomials. Its graph can have vertical asymptotes where the denominator is zero and horizontal or oblique asymptotes describing end behavior. Understanding these features helps interpret the shape and behavior of the graph.
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How to Graph Rational Functions
Solving Equations Using Graphs
To solve an equation like f(x) = k graphically, find the points where the graph of f(x) intersects the horizontal line y = k. The number of intersection points corresponds to the number of solutions to the equation.
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5:47Solving Exponential Equations Using Logs
Interpreting Intersection Points for Solution Count
The number of solutions to f(x) = 3 is the count of x-values where the graph crosses y = 3. Identifying graphs with exactly one intersection with y = 3 is key to answering the question about a single solution.
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Guided course
3:09Example 1
Related Practice
Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation. - ( x +√2)(x-3) < 0
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Textbook Question
Use synthetic division to perform each division. (x5 + 3x4 + 2x3 + 2x2 + 3x+1) / x+2
Textbook Question
Use synthetic division to perform each division. (3x3+6x2-8x+3)/(x+3)
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Textbook Question
Solve each problem. Suppose r varies directly as the square of m, and inversely as s. If r=12 when m=6 and s=4, find r when m=6 and s=20.
Textbook Question
Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x)=(x-k)q(x)+r. ƒ(x)=5x3-3x2+2x-6; k=2