1.03A - Absolute Intro and Basic Function

University:
High School
Course:
High School Physics
Academic year:

2024
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149

Pages:

9
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1 1

ABSOLUTE VALUE FUNCTIONS 1.03A CONCEPT AND BASIC FUNCTION • The absolute value of a real number is the positive value of an expression • The number is placed between vertical lines to signify that we are to ignore its sign. The Concept • 7 =7 • −3 = 3 • 0 =0 𝑥; if 𝑥 is positive • ∀ 𝑥 ∈ ℝ: 𝑥 = ቊ −𝑥; if 𝑥 is negative • If 𝑥 = 7 7 =7 • If 𝑥 = −7 −7 = −(−7) Examples 1. 𝑥 ≥0 • Absolute value is never negative 2. 𝑥 = −𝑥 • Absolute of opposites are equal 3. 𝑥 ∙ 𝑦 = |𝑥| ∙ |𝑦| 4. 𝑥 𝑦 = |𝑥| |𝑦| Properties • Absolute of product equals product of absolutes • Absolute of quotient equals quotient of absolutes. 1. 15 − 56 × 8 + 2. −32 8 −14 2 − 9 × −2 + −4 Try It = 335 = −10 • Textbook: abs 𝑥 Notation: • Read: |𝑥| 𝑓 𝑥 = |𝑥| 𝑥 if 𝑥 ≥ 0 equivalent to 𝑓 𝑥 = ቊ −𝑥 if 𝑥 < 0 *it is a function that can be defined in two parts* The Basic Rule • • • • • • • 𝑑𝑜𝑚 𝑓 = ℝ 𝑟𝑎𝑛 𝑓 = ℝ+ 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 = 0 𝑧𝑒𝑟𝑜 = 0 min 𝑓 = 0 max 𝑓 = 𝐷𝑁𝐸 Variation • • 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ℝ− 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑣𝑒𝑟 ℝ+ • Sign • • 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑣𝑒𝑟 ℝ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟 𝑥 = ∅ • Vertex = (0, 0) • Axis of Symmetry 𝑥 = 0 • Basic Ordered Pairs • { −1, 1 , 0, 0 , 1, 1 } The Basic Properties • HANDOUT: Solving Absolutes Homework