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Understanding Functions and Their Graphs in Calculus

This resource delves into the fundamentals of functions and their graphs, crucial in calculus. It explains the definition of a function, domains, and ranges, distinguishing functions from non-functions through examples. Key concepts such as vertical line tests, even and odd functions, and vertical and horizontal shifts are illustrated with practical exercises. Additionally, it covers function compositions, reinforcing understanding through step-by-step examples. Students will build a solid foundation for analyzing functions in calculus.

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Understanding Functions and Their Graphs in Calculus

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

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

  3. 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.

  4. 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]

  5. 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

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

  7. (2,4) • Find: • the domain • the range • f(-1) = • f(2) = (4,0) [-1,4) [-5,4] -5 (-1,-5) 4 Day 1

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

  9. 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.

  10. 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)

  11. Summary of Graphs of Common Functions f(x) = c y = x y = x 3 y = x2

  12. 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

  13. 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

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

  15. 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

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

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