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Calculus is something to about!!! P.3 Functions and Their Graphs

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Functions Function - for every x there is exactly one y. Domain - set of x-values Range - set of y-values

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Tell whether the equations represent y as a function of x. a.x 2 + y = 1Solve for y. y = 1 – x 2 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 + y 2 = 1 Solve for y. y 2 = x + 1Here we have 2 y’s for each x that we plug in. Therefore, this equation is not a function.

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

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Ex.g(x) = -x 2 + 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

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

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(-1,-5) (2,4) (4,0) Find: a.the domain b.the range c.f(-1) = d.f(2) = [-1,4) [-5,4] -5 4 Day 1

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

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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) A function is y = f(x) is odd if, for each x in the domain of f, f(-x) = -f(x) An even function is symmetric about the y-axis. An odd function is symmetric about the origin.

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Ex. g(x) = x 3 - x g(-x) = (-x) 3 – (-x) = -x 3 + x =-(x 3 – x) Therefore, g(x) is odd because f(-x) = -f(x) Ex. h(x) = x 2 + 1 h(-x) = (-x) 2 + 1 = x 2 + 1 h(x) is even because f(-x) = f(x)

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Summary of Graphs of Common Functions f(x) = c y = x y = x 2 y = x 3

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Vertical and Horizontal Shifts On calculator, graph y = x 2 graph y = x 2 + 2 y = x 2 - 3 y = (x – 1) 2 y = (x + 2) 2 y = -x 2 y = -(x + 3) 2 -1

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Vertical and Horizontal Shifts 1.h(x) = f(x) + cVert. shift up 2.h(x) = f(x) - cVert. 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

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Combinations of Functions The composition of the functions f and g is “f composed by g of x equals f of g of x”

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

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

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