Transcript FUNCTIONS Section 3.1 to 3.3

FUNCTIONS
Section 3.1
RELATIONS
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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:
Y= 3X – 1
If X =2 then Y= 3.2 – 1
So, y=5
Relation
X=2
Input
Y=5
Output
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:
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{(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 = -3
If 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= ± \̸ 1-x2
If x=0 then y = ±1
This 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 – 5
f(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) = \̸ 4-3t
.
Solutions:
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(a) Domain of f is the set of all Real
Numbers.
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(b) Domain of g is {x ̸ x ≠±2}
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(c) Domain of h is { t ̸ t≤4/3}
Tips to find the Domain of a function
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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
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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
10
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f(x)={(1,2);(3,4);(-1,0)}
ANSWER
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Do = { 1, 3, 4}
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ANSWER
f(x) = 2X + 1
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Do= all real numbers
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ANSWER
g(x) =
1
X-1
.
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Do ={ X/ X≠1}
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.
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ANSWER
h(x) = √ X-2
.
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Do={ x/x ≥ 2}
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f(x) = { (1,2); (3,4) ;(-1,0)}
ANSWER
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Rg = { 2, 4, 0}
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ANSWER
f(x) = 2X + 1
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Rg = all real numbers
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ANSWER
g(x) =
1
X-1
.
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Rg = {x/x≠0}
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.
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ANSWER
h(x) = √ X-2
.
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Rg={x/x≥0}