Textbook Question
Use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x) = (2x+7)/(x+3)
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 4, Problem 91
In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. x/(2x+6) − 9/(x2−9)
Verified step by step guidance1
Identify the given expression: .
Factor the denominators where possible: and .
Find the least common denominator (LCD) for the two fractions, which is .
Rewrite each fraction with the LCD as the denominator by multiplying numerator and denominator appropriately: multiply the first fraction by and the second fraction by .
Combine the two fractions into a single fraction by subtracting the numerators over the common denominator, then simplify the numerator algebraically.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Simplifying Rational Expressions
Simplifying rational expressions involves factoring polynomials in the numerator and denominator and reducing common factors. This process makes complex fractions easier to work with and is essential before performing operations like subtraction or addition.
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Simplifying Algebraic Expressions
Operations with Rational Expressions
To subtract rational expressions, you must find a common denominator, rewrite each fraction with this denominator, and then combine the numerators. Understanding how to manipulate denominators and numerators correctly is crucial for accurate simplification.
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Graphing Rational Functions
Graphing rational functions requires identifying key features such as domain restrictions, vertical and horizontal asymptotes, and intercepts. These features help visualize the behavior of the function and are determined from the simplified expression.
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Related Practice
Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. (x - 4)/(x - 1) ≤ 0
Textbook Question
In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. (1 − 3/(x+2)) / (1 + 1/(x−2))
Textbook Question
In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-4) The graph passes through the point (1,4).
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Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. 1/(x + 1) ≤ 2/(x + 4)
Textbook Question
In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. 5x2/(x2−4) ⋅ (x2+4x+4)/(10x3)