315 CHAPTER 2 Test Prep Concepts Examples –3 –3 3 3 –6 6 (x, –y) (x, y) (–x, y) (–x, –y) x2 + y2 = 36 x y 0 x y 0 (1, 2) y = (x – 1)3 + 2 The graph of x2 + y2 = 36 is symmetric with respect to the y-axis, the x-axis, and the origin. The graph of y = 1x - 123 + 2 is the graph of y = x3 translated to the right 1 unit and up 2 units. 2.8 Function Operations and Composition Operations on Functions Given two functions ƒ and g, then for all values of x for which both ƒ1x2 and g1x2 are defined, the following operations are defined. 1 ƒ +g2 1x2 =ƒ1x2 +g1x2 Sum function 1ƒ −g2 1x2 =ƒ1x2 −g1x2 Difference function 1ƒg2 1x2 =ƒ1x2 # g1x2 Product function a ƒ gb 1 x2 = ƒ1x2 g1x2 , g1x2 30 Quotient function The domains of ƒ +g, ƒ −g, and ƒg include all real numbers in the intersection of the domains of ƒ and g, while the domain of ƒ g includes those real numbers in the intersection of the domains of ƒ and g for which g1x2 ≠0. Difference Quotient The line joining P1x, ƒ1x22 and Q1x + h, ƒ1x + h22 has slope m= ƒ1x +h2 −ƒ1x2 h , h 30. The boldface expression is the difference quotient. Let ƒ1x2 = 2x - 4 and g1x2 = 2x. Find each function, and give its domain. 1ƒ + g21x2 = 2x - 4 + 2x 1ƒ - g21x2 = 2x - 4 - 2x t 1ƒg21x2 = 12x - 422x a ƒ gb1x2 = 2x - 4 2x r The domain is 10, ∞2. Let ƒ1x2 = x2 - 5. Find and simplify the expression for the difference quotient. ƒ1x + h2 - ƒ1x2 h = 1x + h22 - 5 - 1x2 - 52 h = x2 + 2xh + h2 - 5 - x2 + 5 h = 2xh + h2 h = h12x + h2 h = 2x + h The domain is 30, ∞2. Reflection across an Axis The graph of y = −ƒ1x2 is the same as the graph of y = ƒ1x2 reflected across the x-axis. The graph of y =ƒ1 −x2 is the same as the graph of y = ƒ1x2 reflected across the y-axis. Symmetry The graph of an equation is symmetric with respect to the y-axis if the replacement of x with -x results in an equivalent equation. The graph of an equation is symmetric with respect to the x-axis if the replacement of y with -y results in an equivalent equation. The graph of an equation is symmetric with respect to the origin if the replacement of both x with -x and y with -y at the same time results in an equivalent equation. Translations Let ƒ be a function and c be a positive number. To Graph: Shift the Graph of y =ƒ1x2 by c Units: y = ƒ1x2 + c up y = ƒ1x2 - c down y = ƒ1x + c2 left y = ƒ1x - c2 right Refer to the graph of x2 + y2 = 36 below. The point 1x, -y2 is the reflection of the point 1x, y2 across the x-axis. The point 1-x, y2 is the reflection of the point 1x, y2 across the y-axis.
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