SECTION 1.1

FUNCTIONS

DEFINITION OF A RELATION

A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, we say that x corresponds to y or that y depends upon x. The correspondence can be written as an ordered pair (x,y).

DEFINITION OF A RELATION

Thus, a relation is simply a set of ordered pairs or a table which relates x and y values.

DEFINITION OF FUNCTION

Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates with each element of X a unique element of Y. This is a special type of relation.

For every x, there is only one y!

DOMAIN AND RANGE

DEFINITION OF DOMAIN The set of all x values.

DEFINITION OF RANGE The set of all y values.

Also called “functional values”.

THE FUNCTION AS A “MAPPING”

x-values y-values 1 4 7 -2 8 2 0 -3 DOMAIN RANGE Ordered Pairs: (1 , 2) (4 , 8) (7, - 3) (- 2, 0)

THE FUNCTION AS A “MAPPING”

Consider 3 students whose names are mapped to their letter grades on the last History exam: Jill Frank Sue A B C For each person in the domain, there can be only one associated letter grade in the range.

-2 -1 0 1 2 3

THE SQUARING FUNCTION

0 1 4 9 Each element in the domain maps to its square.

COUNTER-EXAMPLE:

4 5 1 2 3 Ordered Pairs: (4, 1) (4, 2) (5, 3) This is an example of a relation but not a function.

THREE WAYS TO REPRESENT A FUNCTION

  

NUMERICALLY - ordered pairs SYMBOLICALLY - equation GRAPHICALLY - picture

EXAMPLE

Determine whether the relation represents a function: (a) {(1,4),(2,5),(3,6),(4,7)}

EXAMPLE

Determine whether the relation represents a function: (a) {(1,4),(2,4),(3,5),(6,10)}

EXAMPLE

Determine whether the relation represents a function: (a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)}

EXAMPLE

Determine whether the relation represents a function: (a) {( - 3 ,9),(- 2,4),(0,0),(1,1),( - 3 ,8)}

EVALUATING A FUNCTION AT A GIVEN X-VALUE f(x) = x 2 f(0) = f(2) = 0 4 f(-2) = f(9) = 4 81 x f(x) 0 0 2 4 - 2 4 9 81

Symbolically, the squaring function can be represented as y = x 2 “FUNCTIONAL NOTATION” f(x) = x 2 Read: “f of x equals x squared”

EVALUATING A FUNCTION AT A GIVEN X-VALUE For f(x) = 2x 2 – 3x, find the values of the following: (a) f(3) (b) f(x) + f(3) (d) - f(x) (e) f(x + 3) (f) f(x



h)



f(x) h (c) f(-x)

FINDING VALUES OF A FUNCTION ON A CALCULATOR DO EXAMPLE 7

IMPLICIT FORM OF A FUNCTION When a function is defined by an equation in x and y, we say that the function is given implicitly. If it is possible to solve the equation for y in terms of x, then we write y = f(x) and say that the function is given explicitly. See examples on Pg 127.

DETERMINING WHETHER AN EQUATION IS A FUNCTION Determine if x 2 + y 2 = 1 is a function.

y

 

1



x 2 This means that for certain values of x, there are two possible outcomes for y.

Thus, this is not a function!

1.

2.

Important Facts About Functions: For each x in the domain of a function f , there is one and only one image f(x) in the range. For every x , there is only one y .

f is the symbol we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) f(x) in the range. is another name for y .

3.

Important Facts About Functions:

If y = f(x) , then x is called the independent variable or argument of f , and y is called the dependent variable or the value of f at x .

DOMAIN OF A FUNCTION

If a function is being described symbolically and it comes with a specific domain, that domain should be expressly given.

Otherwise, the domain of the function will be assumed to be the “natural domain”.

EXAMPLE: f(x) = x

2 Knowing the function of squaring a number, we can determine that the natural domain is all real numbers because any real number can be squared.

We can also look at a graph .

EXAMPLE: f(x) = x

2

+ 5x

This is simply a modification of the squaring function. Thus, we can determine that the natural domain is all real numbers.

We can also look at a graph .

Find the domain:

EXAMPLE

f(x) = x 2 3x



4 x

 

2 D: { x



x

 

2 }

Find the domain:

EXAMPLE

f(t) = 4 3t t



4 3

EXAMPLE

Find the domain : f(x) = x 3 + x - 1 All real numbers

OPERATIONS ON FUNCTIONS

Notation for four basic operations on functions: (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (f



g)(x) = f(x)



g(x) (f / g)(x) = f(x) / g(x)

OPERATIONS ON FUNCTIONS

Do Example 10

CONCLUSION OF SECTION 1.1