Domanda del libro di testo
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (4−2x)/(3x+4)≤0
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 4, Problema 51
Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. g(x)=1/(x+2)2
Guida verificata passo dopo passo1
Identify the base function given, which is \( f(x) = \frac{1}{x^2} \). This function has a vertical asymptote at \( x=0 \) and a horizontal asymptote at \( y=0 \).
Recognize that the function \( g(x) = \frac{1}{(x+2)^2} \) is a transformation of \( f(x) = \frac{1}{x^2} \) involving a horizontal shift.
Determine the horizontal shift by setting the inside of the denominator equal to zero: \( x + 2 = 0 \) which gives \( x = -2 \). This means the graph of \( f(x) \) shifts 2 units to the left.
Update the vertical asymptote from \( x=0 \) to \( x = -2 \) because the denominator is zero at \( x = -2 \). The horizontal asymptote remains \( y=0 \) since the degree of the numerator is less than the degree of the denominator.
Sketch the graph by shifting the original \( f(x) = \frac{1}{x^2} \) curve 2 units to the left, keeping the shape the same, with the vertical asymptote at \( x = -2 \) and the horizontal asymptote at \( y=0 \).
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Parent Rational Functions
Parent rational functions like f(x) = 1/x and f(x) = 1/x² serve as the basic models for more complex rational functions. Understanding their graphs, including asymptotes and general shape, is essential before applying transformations. For example, f(x) = 1/x has vertical and horizontal asymptotes at x=0 and y=0, respectively.
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6:04
Intro to Rational Functions
Transformations of Functions
Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For g(x) = 1/(x+2)², the '+2' inside the denominator shifts the graph horizontally left by 2 units. Recognizing how changes inside the function affect the graph helps in sketching the transformed function accurately.
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4:22Domain & Range of Transformed Functions
Asymptotes of Rational Functions
Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, indicating undefined points, while horizontal asymptotes describe end behavior as x approaches infinity. For g(x) = 1/(x+2)², the vertical asymptote is at x = -2, and the horizontal asymptote is y = 0.
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6:24Introduction to Asymptotes
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