By Marcel Finan
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Extra resources for A semester course in trigonometry
4 (a) Using a graphing calculator show that the function f (x) = (b) Now show that f (x) is odd algebraically. 1+x2 x−x3 is odd. Solution. (a) The graph of f (x) is symmetric about the origin so that f (x) is odd. 4. 4 (b) Since f (−x) = 1+(−x)2 (−x)−(−x)3 = 1+x2 −x+x3 = 1+x2 −(x−x3 ) = −f (x) then f (x) is odd. A function can be either even, odd, or neither. 5 (a) Show that the function f (x) = x2 is even but not odd. (b) Show that the function f (x) = x3 is odd but not even. (c) Show that the function f (x) = x + x2 is neither odd nor even.
1 Given f (x) = 3x2 − 1, find (a) f (−4) (b) f 1 3 (c) f (−a) (d) f (x + h) (e) f (x + h) − f (x). 2 x Given f (x) = |x| , find (a) f (4) (b) f (−2) (c) f (x), x > 0 (d) f (x), x < 0. 3 Given f (x) = 3x + 1, if x < 2 −x2 + 11, if x ≥ 2. √ Evaluate: (a) f (−4) (b) f ( 5) (c) f (x), x < 2 (d) f (x + 1), x ≥ 1. 4 Identify the equations that define y as a function of x. (a) 2x + 3y = 7. (b) −x + y 2√ = 2. (c) y = 4 ± x. (d) y 2 = x2 . 5 Identify the collection of ordered pairs (x, y) that define y as a function of x.
So the graph of the new function −f (x) is the reflection of the graph of f (x) about the x-axis. 1 Graph the functions f (x) = 2x and −f (x) = −2x on the same axes. Solution. 1. 1 Reflection About the y-Axis We know that the points (x, f (x)) and (−x, f (x)) are on opposite sides of the y-axis. So the graph of the new function f (−x) is the reflection of the graph of f (x) about the y-axis. 2 Graph the functions f (x) = x3 and f (−x) = −x3 on the same axes. Solution. 2. 2 Symmetry About the y-Axis When constructing the graph of f (−x) sometimes you will find that this new graph is the same as the graph of the original function.
A semester course in trigonometry by Marcel Finan