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# Lesson 1: Functions

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There are many ways we have to represent a function—by a formula, but also by data, by pictures, and by words.

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### Lesson 1: Functions

1. 1. Section 1.1 Functions and their Representations V63.0121.021/041, Calculus I New York University September 7, 2010 Announcements First WebAssign-ments are due September 14 Do the Get-to-Know-You survey for extra credit! . . . . . .
2. 2. Announcements First WebAssign-ments are due September 14 Do the Get-to-Know-You survey for extra credit! . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 2 / 33
3. 3. Function . . . . . .
4. 4. Objectives: Functions and their Representations Understand the definition of function. Work with functions represented in different ways Work with functions defined piecewise over several intervals. Understand and apply the definition of increasing and decreasing function. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 3 / 33
5. 5. What is a function? 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 { y | y = f(x) for some x } is called the range of f. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 4 / 33
6. 6. Outline Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 5 / 33
7. 7. The Modeling Process . . Real-world . . m . odel Mathematical . Problems Model s . olve .est t . i .nterpret . Real-world . Mathematical . Predictions Conclusions . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 6 / 33
8. 8. Plato's Cave . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 7 / 33
9. 9. 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 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 8 / 33
10. 10. Outline Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 9 / 33
11. 11. 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. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 10 / 33
12. 12. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 11 / 33
13. 13. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solution The denominator is zero when x = 2, so the domain is all real numbers except 2. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 11 / 33
14. 14. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solution The denominator is zero when x = 2, so the domain is all real numbers except 2. As for the range, we can solve x+1 2y + 1 y= =⇒ x = x−2 y−1 So as long as y ̸= 1, there is an x associated to y. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 11 / 33
15. 15. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solution The denominator is zero when x = 2, so the domain is all real numbers except 2. As for the range, we can solve x+1 2y + 1 y= =⇒ x = x−2 y−1 So as long as y ̸= 1, there is an x associated to y. Therefore domain(f) = { x | x ̸= 2 } range(f) = { y | y ̸= 1 } . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 11 / 33
16. 16. How did you get that? x+1 start y= x−2 cross-multiply y(x − 2) = x + 1 distribute xy − 2y = x + 1 collect x terms xy − x = 2y + 1 factor x(y − 1) = 2y + 1 2y + 1 divide x= y−1 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 12 / 33
17. 17. 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 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 13 / 33
18. 18. 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. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 14 / 33
19. 19. 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 . . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 14 / 33
20. 20. Functions described numerically We can just describe a function by a table of values, or a diagram. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 15 / 33
21. 21. Example Is this a function? If so, what is the range? x f(x) 1 4 2 5 3 6 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 16 / 33
22. 22. 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 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 16 / 33
23. 23. 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}. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 16 / 33
24. 24. Example Is this a function? If so, what is the range? x f(x) 1 4 2 4 3 6 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 17 / 33
25. 25. 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 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 17 / 33
26. 26. 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}. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 17 / 33
27. 27. Example How about this one? x f(x) 1 4 1 5 3 6 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 18 / 33
28. 28. Example How about this one? . . 1 .. 4 x f(x) 1 4 . .. 2 .. . 5 1 5 3 6 . . 3 .. 6 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 18 / 33
29. 29. 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.” . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 18 / 33
30. 30. An ideal function . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 19 / 33
31. 31. An ideal function Domain is the buttons . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 19 / 33
32. 32. An ideal function Domain is the buttons Range is the kinds of soda that come out . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 19 / 33
33. 33. An ideal function Domain is the buttons Range is the kinds of soda that come out You can press more than one button to get some brands . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 19 / 33
34. 34. An ideal function Domain is the buttons Range is the kinds of soda that come out You can press more than one button to get some brands But each button will only give one brand . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 19 / 33
35. 35. Why numerical functions matter In science, functions are often defined by data. Or, we observe data and assume that it’s close to some nice continuous function. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 20 / 33
36. 36. Numerical Function 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 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 21 / 33
37. 37. Functions described graphically Sometimes all we have is the “picture” of a function, by which we mean, its graph. . . . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 22 / 33
38. 38. 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. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 22 / 33
39. 39. 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. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 23 / 33
40. 40. Outline Modeling Examples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verbally Properties of functions Monotonicity Symmetry . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 24 / 33
41. 41. 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? . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 25 / 33
42. 42. 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 . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 25 / 33
43. 43. 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. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 26 / 33
44. 44. Examples Example Going back to the burrito function, would you call it increasing? . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 27 / 33
45. 45. Examples Example Going back to the burrito function, would you call it increasing? Example Obviously, the temperature in Boise is neither increasing nor decreasing. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 27 / 33
46. 46. 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. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 28 / 33
47. 47. Possible Intensity Graph y . = I(x) . x . . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 29 / 33
48. 48. Possible Gravity Graph y . = F(x) . x . . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 30 / 33
49. 49. 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. . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 31 / 33
50. 50. Examples Even: constants, even powers, cosine Odd: odd powers, sine, tangent Neither: exp, log . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 32 / 33
51. 51. Summary The fundamental unit of investigation in calculus is the function. Functions can have many representations . . . . . . V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 7, 2010 33 / 33