Lesson 1: Functions and their Representations

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Functions can be described with symbols, numbers, words, or pictures.

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Lesson 1: Functions and their Representations

  1. 1. Section 1.1 Functions V63.0121.006/016, Calculus I January 19, 2010 Announcements Syllabus is on the common Blackboard Office Hours TBA . . . . . .
  2. 2. Outline What is a function? Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry . . . . . .
  3. 3. Definition A function f is a relation which assigns to to every element x in a set D a single element f(x) in a set E. The set D is called the domain of f. The set E is called the target of f. The set { f(x) | x ∈ D } is called the range of f. . . . . . .
  4. 4. Outline What is a function? Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry . . . . . .
  5. 5. The Modeling Process . . Real-world . . m . odel Mathematical . Problems Model s . olve .est t . i .nterpret . Real-world . Mathematical . Predictions Conclusions . . . . . .
  6. 6. Plato’s Cave . . . . . .
  7. 7. The Modeling Process . . Real-world . . m . odel Mathematical . Problems Model s . olve .est t . i .nterpret . Real-world . Mathematical . Predictions Conclusions S . hadows F . orms . . . . . .
  8. 8. Outline What is a function? Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry . . . . . .
  9. 9. Functions expressed by formulas Any expression in a single variable x defines a function. In this case, the domain is understood to be the largest set of x which after substitution, give a real number. . . . . . .
  10. 10. Example x+1 Let f(x) = . Find the domain and range of f. x−1 . . . . . .
  11. 11. Example x+1 Let f(x) = . Find the domain and range of f. x−1 Solution The denominator is zero when x = 1, so the domain is all real numbers excepting one. As for the range, we can solve x+1 y+1 y= =⇒ x = x−1 y−1 So as long as y ̸= 1, there is an x associated to y. . . . . . .
  12. 12. No-no’s for expressions Cannot have zero in the denominator of an expression Cannot have a negative number under an even root (e.g., square root) Cannot have the logarithm of a negative number . . . . . .
  13. 13. Piecewise-defined functions Example Let { x2 0 ≤ x ≤ 1; f(x) = 3−x 1 < x ≤ 2. Find the domain and range of f and graph the function. . . . . . .
  14. 14. Piecewise-defined functions Example Let { x2 0 ≤ x ≤ 1; f(x) = 3−x 1 < x ≤ 2. Find the domain and range of f and graph the function. Solution The domain is [0, 2]. The range is [0, 2). The graph is piecewise. . . 2 . . . 1 . . . . . 0 . 1 . 2 . . . . . . .
  15. 15. Functions described numerically We can just describe a function by a table of values, or a diagram. . . . . . .
  16. 16. Example Is this a function? If so, what is the range? x f(x) 1 4 2 5 3 6 . . . . . .
  17. 17. Example Is this a function? If so, what is the range? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 2 5 3 6 . . 3 .. 6 . . . . . .
  18. 18. Example Is this a function? If so, what is the range? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 2 5 3 6 . . 3 .. 6 Yes, the range is {4, 5, 6}. . . . . . .
  19. 19. Example Is this a function? If so, what is the range? x f(x) 1 4 2 4 3 6 . . . . . .
  20. 20. Example Is this a function? If so, what is the range? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 2 4 3 6 . . 3 .. 6 . . . . . .
  21. 21. Example Is this a function? If so, what is the range? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 2 4 3 6 . . 3 .. 6 Yes, the range is {4, 6}. . . . . . .
  22. 22. Example How about this one? x f(x) 1 4 1 5 3 6 . . . . . .
  23. 23. Example How about this one? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 1 5 3 6 . . 3 .. 6 . . . . . .
  24. 24. Example How about this one? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 1 5 3 6 . . 3 .. 6 No, that one’s not “deterministic.” . . . . . .
  25. 25. In science, functions are often defined by data. Or, we observe data and assume that it’s close to some nice continuous function. . . . . . .
  26. 26. Example Here is the temperature in Boise, Idaho measured in 15-minute intervals over the period August 22–29, 2008. . 1 . 00 . 9 .0. 8 .0. 7 .0. 6 .0. 5 .0. 4 .0. 3 .0. 2 .0. 1 .0. . . . . . . . 8 . /22 . /23 . /24 . /25 . /26 . /27 . /28 . /29 8 8 8 8 8 8 8 . . . . . .
  27. 27. Functions described graphically Sometimes all we have is the “picture” of a function, by which we mean, its graph. . . . . . . . .
  28. 28. Functions described graphically Sometimes all we have is the “picture” of a function, by which we mean, its graph. . . The one on the right is a relation but not a function. . . . . . .
  29. 29. Functions described verbally Oftentimes our functions come out of nature and have verbal descriptions: The temperature T(t) in this room at time t. The elevation h(θ) of the point on the equator at longitude θ. The utility u(x) I derive by consuming x burritos. . . . . . .
  30. 30. Outline What is a function? Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry . . . . . .
  31. 31. Monotonicity Example Let P(x) be the probability that my income was at least $x last year. What might a graph of P(x) look like? . . . . . .
  32. 32. Monotonicity Example Let P(x) be the probability that my income was at least $x last year. What might a graph of P(x) look like? . . 1 . .5 . 0 . . . $ .0 $ . 52,115 $ . 100K . . . . . .
  33. 33. Monotonicity Definition A function f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f. A function f is increasing if f(x1 ) < f(x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f. . . . . . .
  34. 34. Examples Example Going back to the burrito function, would you call it increasing? . . . . . .
  35. 35. Examples Example Going back to the burrito function, would you call it increasing? Example Obviously, the temperature in Boise is neither increasing nor decreasing. . . . . . .
  36. 36. Symmetry Example Let I(x) be the intensity of light x distance from a point. Example Let F(x) be the gravitational force at a point x distance from a black hole. . . . . . .
  37. 37. Possible Intensity Graph y . = I(x) . x . . . . . . .
  38. 38. Possible Gravity Graph y . = F(x) . x . . . . . . .
  39. 39. Definitions Definition A function f is called even if f(−x) = f(x) for all x in the domain of f. A function f is called odd if f(−x) = −f(x) for all x in the domain of f. . . . . . .
  40. 40. Examples Even: constants, even powers, cosine Odd: odd powers, sine, tangent Neither: exp, log . . . . . .

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