advertisement

Math 211-330/332 Worksheet 1 Jan 20, 2016 Solutions Review: • Definitions: function, domain, range, independent variable, dependent variable; • Think of a function as a machine (I prefer “vending machine”); • Representations of functions: words, table, graph, mathematical formula; • How to determine the domain of a function; • The vertical line test. 1. Some of these algebraic manipulations are correct, and others are incorrect. (Note that a correct answer might be obtained by using incorrect reasoning.) Identify the correct ones and fix the incorrect ones: (a) (x + y)2 x2 + y 2 = = x2 + y. y y Solution. This is incorrect. (x + y)2 x2 + 2xy + y 2 x2 Correct version: = = + 2x + y. y y y Ç å2 Å ã2 x2 1 (x2 )3 2 = x · = x2 · (b) 3 2 3 (x ) x x Solution. This is correct. (c) = x2 = 1. x2 x2 1 x2 − 1 = − = x − 1. x+1 x 1 Solution. This is incorrect. x2 − 1 (x − 1)(x + 1) Correct version: = = x − 1. x+1 x+1 Å ã 2−4 −4 −2 4 1 (d) −2 = = = . 2 −2 16 4 Solution. This is incorrect. 2−4 22 1 1 Correct version: −2 = 4 = 2 = . 2 2 2 4 Å (e) x−1 + y −1 (x + y)−1 x+y = = −1 −1 x −y (x − y)−1 x−y Solution. This is incorrect. Correct version: ã−1 =− x+y x+y = . x−y x−y y+x 1 1 + y+x x−1 + y −1 x y xy = y−x = = . −1 −1 1 1 x −y y−x − xy x y 2. Determine the domain of each function √ (a) f (x) = x2 + 1 (b) f (x) = 1 x2 −3x+2 Solution. (a) x2 + 1 ≥ 0 ⇒ x ∈ (−∞, +∞); (b) x2 − 3x + 2 6= 0 ⇒ (x − 1)(x − 2) 6= 0 ⇒ x 6= 1, 2 3. If f (x) = x2 − 3x + 2, evaluate (a) f (1) (b) f (t) (c) f (x + 1) (d) f (1+t)−f (1) t Solution. (a) f (1) = 12 − 3 ∗ 1 + 2 = 0 (b) f (t) = t2 − 3t + 2 (c) f (x + 1) = (x + 1)2 − 3(x + 1) + 2 (d) f (1+t)−f (1) t = [(1+t)2 −3(1+t)+2]−0 t = 1+2t+t2 −3−3t+2 t = t2 −t t = t(t−1) t =t−1