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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 91
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)

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Identify the given expression: x2x+6 - 9x2 - 9.
Factor the denominators where possible: 2x + 6 = 2(x + 3) and x2 - 9 = (x - 3)(x + 3).
Find the least common denominator (LCD) for the two fractions, which is 2(x - 3)(x + 3).
Rewrite each fraction with the LCD as the denominator by multiplying numerator and denominator appropriately: multiply the first fraction by (x - 3)/(x - 3) and the second fraction by 2/2.
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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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

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)

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Textbook Question

Solve each inequality in Exercises 86–91 using a graphing utility. (x - 4)/(x - 1) ≤ 0

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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))

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Textbook Question

Solve each inequality in Exercises 86–91 using a graphing utility. 1/(x + 1) ≤ 2/(x + 4)

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Textbook Question

Solve each inequality in Exercises 86–91 using a graphing utility. x3 + x2 - 4x - 4 > 0

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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)

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