Let A and B be two sets. For all or some of the elements of set A, joined by some sort of condition or property to set B, we are telling about a relation.

Presentación del tema: "Let A and B be two sets. For all or some of the elements of set A, joined by some sort of condition or property to set B, we are telling about a relation."— Transcripción de la presentación:

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2 Let A and B be two sets. For all or some of the elements of set A, joined by some sort of condition or property to set B, we are telling about a relation (R) between the elements of set A and B. (Fuenlabrada, 2001) (1, 4)(1,5) (2,4)(2,5) (3,4)(3,5) 123123 4545 x y A B

3 Domain: The set of the first series values belonging to the relation. (A) D = {1,2,3} Range: The set of the second series of values belonging to the relation. (B) R = {4,5} (1, 4)(1,5) (2,4)(2,5) (3,4)(3,5) 123123 4545 x y A B Domain Range

4 They are a particular case of relations. It is a relationship between two variables such that to each value of the independent variable there corresponds exactly one value of the dependent variable. (Larson, et al. 1995) (1, 4) (2,5) (3,5) 123123 4545 x y A B X independent variable Y dependent variable

5 Domain: The set of the first series values belonging to the relation. (A) D = {1,2,3} Range: The set of the second series of values belonging to the relation. (B) R = {4,5} x y Domain Range (1, 4) (2,5) (3,5) 123123 4545 A B X independent variable Y dependent variable

6 Injective function: Each element of the range matches with just one element of the domain. There can be remaining elements in the range. Surjective function: Each element of the range matches with one or more elements of the domain. Bijective function: Each element of the range matches with only one element of the domain.

7 A B One to one A B One to many Function Relation A B Many to one Function A B Many to many Relation

8 It is a function when plotting the values there is only one point vertically speaking. y=x Domain = { R } Range = { R } yx 11 22 33 44 55 ……

9 Some other examples of functions and relations. 1=x 2 +y 2 Relation y=(x 2 -1) 2 Function y=-x-6 Function x 2 +y=1 Function x+y 2 =1 Relation y=1 Function

10 Determine if the following set of pairs is a function or a relation: 1)A={-2,-1,0,1,2} B={1,2} AxB= (-2,1) (-1,1) (0,1) (1,1) (2,1) (-2,2) (-1,2) (0,2) (1,2) (2,2) Plot the graph, determine domain and range. 2)A={-1,0,4,6} B={3,4} AxB= Determine Plot the Graph, determine domain and range. 3)y=4x+5; Determine some points, the domain, range and plot the graph. 4) ; Determine some points, the domain, range and plot the graph. 5) ; Determine some points, the domain, range and plot the graph.

11 So then, a relation becomes a function when: -All elements contained in the domain are related to one element in the range. -There must not exist any element in the domain that relates to more than one element belonging to the range. -There can exist elements in the range that matches to more than one element in the domain.

12 Several common functions have graphs that are symmetric to the y-axis or the origin. If the graph of f is symmetric to the y-axis; then: f(-x) = f(x) And f is called an even function. If the graph of f is symmetric to the origin, then: f(-x) = -f(x) And f is called an odd function.

13 Even Functions:Odd Functions: Plot these graphs using some points, find the domain and range.

14 But, how do we prove that a function is odd or even? When proving a function even, we have to substitute x by –x. The result have to be the same. E.g. So, here we have the same result, this means that the function is even. When proving a function odd, we have to follow the same procedure. The result must be symmetric. E.g. This shows us that the function is Odd.

15 Even or Odd?

16 Let f be a one to one function with domain D and range R. If g is a function with domain R and range D, then g is the inverse function of f if and only if both of the following conditions are true: 1.g(f(x)) = x for every x in D 2.f(g(y)) = y for every y in R. a)First f, then g : b) First g, then f: f is a function from D to R g is a function from R to D Domain of f -1 = Range of f Range of f -1 = Domain of f (Larson, et al. 1995) xf(x) Domain (D) Range (R) g f g(f(x)) g(y) y Domain (D) Range (R) g f f(g(y))

17 In order to check if a function is inverse we have the primary function: Now we find the inverse in this way: Watch that we changed the x by y and vice versa. Now we begin substituting; the result must be the same. Substituting (f o f -1 )Substituting (f -1 o f)

18 Now, try for yourself, make both the graph for the original function and the inverse: 1. 2. 3.

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20 The exponential function f with base a is denoted by, where, and x is any real number. And why ? Because: We would obtain a horizontal line. Domain = { R } Range = (0, )

21 Domain = { R } Range = (0, ) e.g. Graph of exponential Growth Domain = { R } Range = (0, ) e.g. Graph of radioactive decay Domain = { R } Range = (-, 0) Domain = { R } Range = (-, 0)

22 Horizontal Shifting: Domain = { R } Range = (0, ) Domain = { R } Range = (0, )

23 Horizontal Shifting (Part 2): f(x)=3 -x f(x)=3 (-x-2)

24 Vertical Shifting: Domain = { R } Range = (0, ) Domain = { R } Range = (-2, )

25 Bell Shaped Graph: Domain = { R } Range = (0, 1]

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27 Application: Bacterial growth Exponential functions may be used to describe the growth of certain populations. As an illustration, suppose it is observed experimentally that the number of bacteria in a culture doubles every day. If 1000 bacteria are present at the start, then we obtain the following table, where t is the time in days and f(t) is the bacteria count in time t. t (time in days) 01234 f(t)100020004000800016000 It appears that. With this formula we can predict the number of bacteria present at any time t. For example at

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29 Drug Dosage: A drug is eliminated from the body through urine. Suppose that from an initial dose of 10 mg, the amount A(t) in the body t hours later is given by: a)Estimate the amount of the drug in the body 8 hours after the initial dose. b)What percentage of the drug is eliminated each hour?

30 You can also apply logarithms to calculate: Earthquakes Cooling systems Population doubling time Half-life of a radioactive substance Sound Intensity Electrical circuits… And much more.

31 Let a be a positive real number different from 1. The Logarithm of x with base a is defined by: if and only if for every x>0 and for every real number y. Logarithmic formExponential form

32 So, we can conclude that an Logarithmic function is the inverse of an exponential one:

33 Horizontal Shifting: Domain = (0, ) Range = { R } Domain = (2, ) Range = { R }

34 Domain = (0, ) Range = { R } Domain = (0, ) Range = { R } Vertical Shifting:

35 Transform the following exponential forms into logarithmic ones and vice versa. Remember: Logarithmic formExponential form

36 Property of Logarithms

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38 An absolute value never allows a number to be negative; so: Domain = { R } Range = [0, ) Domain = { R } Range = (-, 0]

39 Vertical Shifting: Domain = { R } Range = [2, ) Domain = { R } Range = [-2, )

40 Odd powers change to a quadratic form: Domain = { R } Range = [0, ) Domain = { R } Range = [0, )

41 Asymptotic graphs receive only positive values, nevertheless the asymptote remains: Domain = { R 0 } Range = (0, ) Some Exercises: 1.4. 2.5. 3.6.

42 When plotting a trigonometric graph make sure to change your x axis to radians. Sine function Domain = { R } Range = [-1, 1] Cycle starts at 0, finishes at 2  Cosine function Domain = { R } Range = [-1, 1] Cycle starts at 1, finishes at 2  Tangent function Cycle starts at 0, finishes at 2  Asymptotes each odd  divided by two

43 Inverse trigonometric functions are plotted on the opposite sides, lets see: Cosecant function Cycle starts at 0, finishes at 2  When writing the range consider the gap. Secant function Cycle starts at 1, finishes at 2  Cotangent function Cycle starts at 0, finishes at 2  Asymptotes each odd  divided by two CSC sin

44 Functions are sometimes described by more than one expression: Determine Domain and Range:

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46 -

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48 l 1. 2. 3. 4. 5.

49 1)To find the x-intercepts: equal the numerator to 0. 2)To find the vertical asymptotes: equal the denominator to 0. 3)To find the y-intercepts: evaluate the whole function for x=0. The result will be (0,y) 4)To find the horizontal asymptotes: a.If n < m, then the x-axis (the line y=0) is the horizontal asymptote for the graph or f. b.If n = m, then the line y= a/b is the horizontal asymptote for the graph of f. c.If n > m, the graph of f has no horizontal asymptote. 5) The vertical asymptotes divides the xy-plane into regions. Substitute the values In order to plot these graphs there is a method of the following simple steps:

50 1. 2. 3. 4. 5. 6. 7. Find every element of the rational function to plot them. Finally find domain and range.

51 Composition of functions: The function given by (f o g)x = f(g(x)) is the composite of f with g. The domain of (f o g) is the set of all x in the domain of g such that g(x) is the domain of f. (Larson, et al. 1995) For Example: and a.f(g(x)) b. g(f(x))

52 Lets make the following operations:

53 Try for yourself: 1. a.(f o f)x b.(g o g)x c.(f o g)x d.(g o f)x 2. a.(f o g)x b.(g o f)x

54 For example: Addition: Subtraction: Multiplication: Division: As easy as that !!!

55 Try for yourself: 1.(f – h – g)x 2.(f / k)x 3.((f / g) + h)x 4.((g / h) – f )x