advertisement

Calculus is something to P.3 Functions and Their Graphs about!!! Functions Function - for every x there is exactly one y. Domain - set of x-values Range - set of y-values Tell whether the equations represent y as a function of x. a. x2 + y = 1 Solve for y. y = 1 – x2 For every number we plug in for x, do we get more than one y out? No, so this equation is a function. b. -x + y2 = 1 Solve for y. y2 = x + 1 Here we have 2 y’s for each x that we plug in. Therefore, this equation is not a function. y 1 x Find the domain of each function. a. f: {(-3,0), (-1,4), (0,2), (2,2), (4,-1)} Domain = { -3, -1, 0, 2, 4} b. c. 1 g ( x) x5 f ( x) 4 x D: 2 (2 x)(2 x) 0 D: [-2, 2] , x 5 Set 4 – x2 greater than or = to 0, then factor, find C.N.’s and test each interval. Ex. g(x) = -x2 + 4x + 1 Find: a. g(2) b. g(t) c. g(x+2) d. g(x + h) Ex. x 1, x 0 f ( x) x 1, x 0 2 Evaluate at x = -1, 0, 1 Ans. 2, -1, 0 Ex. f(x) = x2 – 4x + 7 Find. f ( x h) f ( x ) h [(x h) 4( x h) 7] [ x 4 x 7] h 2 2 2 x 2 xh h 4 x 4h 7 x 4 x 7 h 2 h(2 x h 4) 2 xh h 4h = 2x + h - 4 h h 2 2 (2,4) Find: a. the domain [-1,4) b. the range [-5,4] c. f(-1) = -5 (4,0) (-1,-5) d. f(2) = 4 Day 1 Vertical Line Test for Functions Do the graphs represent y as a function of x? no yes yes Tests for Even and Odd Functions A function is y = f(x) is even if, for each x in the domain of f, f(-x) = f(x) An even function is symmetric about the y-axis. A function is y = f(x) is odd if, for each x in the domain of f, f(-x) = -f(x) An odd function is symmetric about the origin. Ex. g(x) = x3 - x g(-x) = (-x)3 – (-x) = -x3 + x = -(x3 – x) Therefore, g(x) is odd because f(-x) = -f(x) Ex. h(x) = x2 + 1 h(-x) = (-x)2 + 1 = x2 + 1 h(x) is even because f(-x) = f(x) Summary of Graphs of Common Functions f(x) = c y=x y x y x y=x3 y = x2 Vertical and Horizontal Shifts On calculator, graph y = x2 graph y = x2 + 2 y = x2 - 3 y = (x – 1)2 y = (x + 2)2 y = -x2 y = -(x + 3)2 -1 Vertical and Horizontal Shifts 1. h(x) = f(x) + c Vert. shift up 2. h(x) = f(x) - c Vert. shift down 3. h(x) = f(x – c) Horiz. shift right 4. h(x) = f(x + c) Horiz. shift left 5. h(x) = -f(x) Reflection in the x-axis 6. h(x) = f(-x) Reflection in the y-axis Combinations of Functions The composition of the functions f and g is ( f g )(x) f ( g ( x)) “f composed by g of x equals f of g of x” Ex. f(x) = g(x) = x - 1 x f g (2) f g x f ( g ( x)) Find of 2 x 1 2 1 1 Ex. f(x) = x + 2 and g(x) = 4 – x2 f g x Find: f(g(x)) = (4 – x2) + 2 = -x2 + 6 g f x g(f(x)) = 4 – (x + 2)2 = 4 – (x2 + 4x + 4) = -x2 – 4x 1 Ex. Express h(x) = as a composition 2 x 2 of two functions f and g. f(x) = 1 2 x g(x) = x-2