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. Solve for y. b. -x + y2 = 1 y2 = x + 1 Here we have 2 y’s for each x that we plug in. Therefore, this equation is not a function.

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. D: Set 4 – x2 greater than or = to 0, then factor, find C.N.’s and test each interval. c. D: [-2, 2]

Ex. g(x) = -x2 + 4x + 1 Find: a. g(2) b. g(t) c. g(x+2) d. g(x + h) Ex. Evaluate at x = -1, 0, 1 Ans. 2, -1, 0

Ex. f(x) = x2 – 4x + 7 Find. = 2x + h - 4

(2,4) Find: the domain the range f(-1) = f(2) = (4,0) [-1,4) [-5,4] -5 (-1,-5) 4 Day 1

Vertical Line Test for Functions Do the graphs represent y as a function of x? yes yes no

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 3 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 composed by g of x equals f of g of x”

Ex. f(x) = g(x) = x - 1 Find of 2 Ex. f(x) = x + 2 and g(x) = 4 – x2 Find: f(g(x)) = (4 – x2) + 2 = -x2 + 6 g(f(x)) = 4 – (x + 2)2 = 4 – (x2 + 4x + 4) = -x2 – 4x

Ex. Express h(x) = as a composition of two functions f and g. f(x) = g(x) = x - 2