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 49
Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. g(x)=1/(x+1) − 2
Guida verificata passo dopo passo1
Identify the base function given, which is \( f(x) = \frac{1}{x} \). This is the parent function for the transformation.
Recognize the transformation inside the function's argument: \( x + 1 \). This represents a horizontal shift to the left by 1 unit because adding 1 inside the function moves the graph left.
Next, observe the \( -2 \) outside the fraction. This indicates a vertical shift downward by 2 units.
Combine these transformations to write the transformed function as \( g(x) = \frac{1}{x + 1} - 2 \). This means the graph of \( f(x) = \frac{1}{x} \) is shifted left 1 unit and down 2 units.
To graph \( g(x) \), start with the graph of \( f(x) = \frac{1}{x} \), shift every point left by 1, then shift every point down by 2. Also, update the vertical and horizontal asymptotes accordingly: the vertical asymptote moves from \( x=0 \) to \( x=-1 \), and the horizontal asymptote moves from \( y=0 \) to \( y=-2 \).
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Parent Rational Functions
The parent functions f(x) = 1/x and f(x) = 1/x² are basic rational functions with distinct shapes and asymptotes. Understanding their graphs, including vertical and horizontal asymptotes, is essential as they serve as the starting point for transformations.
Video consigliato:
6:04
Intro to Rational Functions
Transformations of Functions
Transformations include shifts, stretches, and reflections applied to the parent function. For g(x) = 1/(x+1) - 2, the graph shifts left by 1 unit and down by 2 units, altering the position of asymptotes and the overall graph without changing its shape.
Video consigliato:
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, and horizontal asymptotes describe end behavior. Transformations shift these asymptotes accordingly, crucial for accurate graphing.
Video consigliato:
6:24Introduction to Asymptotes
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