Textbook Question
Determine whether each relation defines a function, and give the domain and range. See Examples 1 – 4.
Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem 27
Write each rational expression in lowest terms. See Example 2. (m² - 4m + 4) / (m² + m - 6)
Verified step by step guidance1
Start by factoring both the numerator and the denominator of the rational expression separately. For the numerator \(m^{2} - 4m + 4\), look for two numbers that multiply to \(4\) and add to \(-4\).
Factor the numerator as a perfect square trinomial: \(m^{2} - 4m + 4 = (m - 2)^{2}\).
Next, factor the denominator \(m^{2} + m - 6\). Find two numbers that multiply to \(-6\) and add to \(1\).
Factor the denominator as \((m + 3)(m - 2)\).
Write the rational expression as \(\frac{(m - 2)^{2}}{(m + 3)(m - 2)}\) and then simplify by canceling the common factor \((m - 2)\), leaving the expression in lowest terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Quadratic Expressions
Factoring involves rewriting a quadratic expression as a product of two binomials. For example, m² - 4m + 4 factors to (m - 2)(m - 2). Recognizing patterns like perfect square trinomials or factoring by grouping helps simplify expressions.
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Factoring
Simplifying Rational Expressions
A rational expression is simplified by factoring the numerator and denominator and then canceling common factors. This reduces the expression to its lowest terms, making it easier to work with or evaluate.
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2:58Rationalizing Denominators
Identifying and Excluding Restrictions
When simplifying rational expressions, it's important to identify values that make the denominator zero, as these are excluded from the domain. This ensures the expression remains defined and valid.
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3:43Finding the Domain of an Equation
Related Practice
Textbook Question
Solve each linear equation. See Examples 1–3. 0.2x - 0.5 = 0.1x + 7
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Textbook Question
Concept Check Graph the points on a coordinate system and identify the quadrant or axis for each point. (4.5, 7)
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Textbook Question
Determine whether each relation defines a function, and give the domain and range. See Examples 1 – 4.
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Textbook Question
Write each rational expression in lowest terms. See Example 2. (8m² + 6m - 9) / (16m² - 9)
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Textbook Question
Solve each linear equation. See Examples 1–3.
(3x - 1)/4 + (x + 3)/6 = 3