GENERAL MATHEMATICS Module 1: Review on Functions

REVIEW ON FUNCTIONS

Module 1

GENERAL MATHEMATICS Samar College Galina V. Panela


What is a function?

Definition of a Function:

• It is a relation define as a set of ordered pairs (x, y) where no two or more distinct ordered pairs have the same first element (x).

• Every value of x corresponds to a unique value of y


Examples:• Illustrations below are examples of a

function


Is it a function or not?



What is the difference between a function and a

relation?

RELATIONS versus FUNCTIONS


RELATIONS FUNCTIONSA relation is a rule that relates values from a set of values called the domain to a second set of values called the range.

A function is a relation where each element in the domain is related to only one value in the range by some rule.



RELATIONS FUNCTIONSThe elements of the domain can be imagines as input to a machine that applies rule to these inputs to generate one or more outputs.

The elements of the domain can be imagined as input to a machine that applies a rule so that each input corresponds to only one output.



RELATIONS FUNCTIONS

A relation is also a set of ordered pairs (x, y).

A function is a set of ordered pairs (x, y) such that no two ordered pairs have the same x-value but different y-values.


a. f = {(0, -1), (2, -5), (4, -9), (6,-13)}b. r ={(a, 0), (b, -1), (c, 0), (d, -1)}c. g = (5, -10), (25, -50), (50, -100)d. t = {(-2, 0), (-1, 1), (0, 1), (-2, 2)}


The function as a machine…

We will try to represent mathematical relations as machines with an input and an output, and that the output is related to the input by some rule.


Determine if this machine produces a function…









What is a table of values?

Table of Valuesa

• A table of values is commonly observed when describing a function.

• This shows the correspondence between a set of values of x and a set of values of y in a tabular form.


Examples of Table of Valuesa


x 0 1 4 9 16y - 5 - 4 - 1 4 11

x -1 -1/4 0 1/4 1y -1 - 1/2 0 1/2 1


1.A jeepney and its plate number2.A student and his ID number3.A teacher and his cellular phone4.A pen and the color of its ink



What is a vertical line test?

Vertical Line Testa

• The vertical line test for a function states that if each vertical line intersects a graph in the x-y plane at exactly one point, then the graph illustrates a function.


Is this a function or not?






Relationship Between the Independent and Dependent Variables


Input (value of x)

Process (equation

rule)Output

(value of y)

Examples:

1. Find the value of y in the equation y = 10x – 3 if x = - 5.

2. Find the value of x if the value of y in the equation is 2.


Applications:1. A car has travelled a distance of 124

kilometers in 4 hours. Find the speed of the car.

2. The volume of the cube is defined by the function where s is the length of the edge. • What is the volume of the cube if the

length of the edge is 5 cm?• What is the length of its edge if its

volume is 728 cubic meters?


REVIEW ON FUNCTIONSModule 1


EVALUATING FUNCTIONS

Evaluating Functions a

• It is the process of determining the value of the function at the number assigned to a given variable.


Example:Let . Find the following values of the function

a. f (2)b. f (-1)c. f (0)d. f (- ½ )e. f (- 4)


Example:Let . Find the following values of the function

a. g (2)b. g (4)c. g (0)d. g (9)e. g (- 1/3)


Example:Let h. Find the following values of the function

a. h (1)b. h (-2)c. h (6)d. h (0)e. h (2)




DOMAIN AND RANGE OF FUNCTIONS

Domain D of a Functiona

• It is the set of all x-coordinates in the set of ordered pairs.

Range R of a Functiona

• It is the set of all y-coordinates in the set of ordered pairs.


Determine the domain and the range of the following:a


x 0 1 4 9 16y - 5 - 4 - 1 4 11

x -1 -1/4 0 1/4 1y -1 - 1/2 0 1/2 1

More on Independent Variablesa

• There are instances in which not all values of the independent variables are permissible.

• That is, some functions have restrictions.


Determine the domain and the range of the following:

a


Piece-wise Functionsa

• These are functions which are defined in defined in different domains since they are determined by several equations.


Determine the domain and the range of the following:

a


{ 2x + 3 if x ≠ 2 4 if x = 2

{ 2x + 3 if x < 1– if x 1



OPERATIONS ON FUNCTIONS

Operations on Functions

If f and g are functions then• (f + g) = f(x) + g(x)• (f – g) = f(x)– g(x)• (f g) = f(x) g(x)• where g(x) ≠ 0


ExampleLet f(x) = and g(x)= x – 1. Perform the operations and identify the domain• (f + g)• (f – g)• (f g)


ExampleLet f(x)= x – 3 and g(x) = . Perform the operations and identify the domain• (f + g)• (f – g)• (f g)




COMPOSITE FUNCTIONS


If f and g are functions then the composite function denoted by , is defined by



The domain of is the set of all numbers x in the domain of g such that g(x) is in the domain of f.


Example

Let f(x)= x – 3 and g(x) = . Find • )(x)• )(x)• )(3)• )(- 4)




EVEN AND ODD FUNCTIONS

Even and Odd Functions

• A function f is said to be even if f(–x)=f(x) for each value of x in the domain of f.

• A function f is said to be odd if f(–x)= – f(x) for each value of x in the domain of f.


ExampleDetermine whether each of the following functions is even, odd or neither


Education

GENERAL MATHEMATICS Module 1: Review on Functions