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) x y A B

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) x y A B Domain Range

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) x y A B X independent variable Y dependent variable

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) A B X independent variable Y dependent variable

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.

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

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

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

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.

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.

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.

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

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.

Even or Odd?

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))

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)

Now, try for yourself, make both the graph for the original function and the inverse:

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, )

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)

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

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

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

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

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) f(t) It appears that. With this formula we can predict the number of bacteria present at any time t. For example at

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?

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.

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

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

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

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

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

Property of Logarithms

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

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

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

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

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

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

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

-

+ 2

l

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:

Find every element of the rational function to plot them. Finally find domain and range.

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))

Lets make the following operations:

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

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

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