FUNCTIONSFUNCTIONSSection 3.1 Section 3.1

RELATIONS

Definition: 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→y, then we say that

“x” correspond to “y” or that “y” depends on “x”

Example:

Relation

X=2

Input Y=5Output

Y= 3X – 1If X =2 then Y= 3.2 – 1So, y=5

A map: illustrate a relation by using a set of inputs and drawing arrows to the corresponding elements in the set of outputs.

Ordered pairs can be used to represent x→y as ( x , y )

{ (0,-2),(0,1),(1,2),(2,1), (3,4)}

Determine whether a relation represents a Function

Let X and Y be two nonempty sets. “ A function from X into Y is a relation that associates with each element of X exactly one element of Y”

Let’s now use ordered pairs to identify which of these sets are relations or functions:

{(1,4) , (2,5) , (3,6), (4,7)} Do={1,2,3,4} Rg={4,5,6,7}

{(1,4),(2,4)(3,5),(6,10)} Do={1,2,3,6} Rg={4,5,10}

{ (-3,9), (-2,4), (0,0), (1,1), (-3,8)}

Determine whether an Equation is a Function

Determine if the equation y=2x – 5 defines y as a function of x

If x=1, then y=2(1) – 5 = -3If x=3, then y= 2(3) – 5 = 1

The equation is a FUNCTION

Example 2

Determine if the equation x2+y2=1 defines y as a function of x.

Solve for y: y2= 1 - x2

y= ± \̸B 1-x2

If x=0 then y = ±1This means the equation x2+y2=1

does not define a function

Find the value of a Function

y = f(x) read “f of x”

Example:y=f(x) = 2x – 5 then f(1/2)=2.1/2 – 5f(1/2)= -4

The variable x is called independent variable or argument, and y is called dependent variable

Finding the Domain of a Function

The domain of a function is the largest set of Real numbers for which the value f(x) is a Real number.

Examples:Find the domain of each of the following

functions:(a) f(x)= x2+5x (b) g(x)= 3x .

x2-4 (c) h(t) = \̸B 4-3t

Solutions:

(a) Domain of f is the set of all Real Numbers.

(b) Domain of g is {x B x ≠±2}

(c) Domain of h is { t B t≤4/3}

Tips to find the Domain of a function

Start with the domain as the set of real numbers.

If the equation has a denominator, exclude any numbers that give a zero denominator.

If the equation has a radical of even index, exclude any numbers that cause the expression inside the radical to be negative.

SUMMARY Function: a relation between two sets of real

numbers so that each number x in the first set, the domain, has corresponding to it exactly one number y in the second set, the range.

Unspecified Domain: If a function f is defined by an equation and no domain is specified, then the domain will be taken to be the largest set

of real numbers for which the equation defines a real number.

Function Notation: y= f(x)f is the symbol for the variable, x is the

independent variable or argument, y is the dependent variable, and f(x) is the value of the

function at x, or the image of x.

GAME TIME

DOMAIN RANGE

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2020

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40 40

f(x)={(1,2);(3,4);(-1,0)}

ANSWER

Do = { 1, 3, 4}

f(x) = 2X + 1

ANSWER

Do= all real numbers

g(x) = 1 .

X - 1

ANSWER

Do ={ X/ X≠1}

. .

h(x) = √ X-2

ANSWER

Do={ x/x ≥ 2}

f(x) = { (1,2); (3,4) ;(-1,0)}

ANSWER

Rg = { 2, 4, 0}

f(x) = 2X + 1

ANSWER

Rg = all real numbers

g(x) = 1 .

X - 1

ANSWER

Rg = {x/x≠0}

. .

h(x) = √ X-2

ANSWER

Rg={x/x≥0}