FUNCTIONS Section 3.1
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Transcript of FUNCTIONS Section 3.1
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FUNCTIONSFUNCTIONSSection 3.1 Section 3.1
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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”
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Example:
Relation
X=2
Input Y=5Output
Y= 3X – 1If X =2 then Y= 3.2 – 1So, y=5
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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)}
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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”
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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)}
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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
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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
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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
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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
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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}
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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.
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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.
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GAME TIME
DOMAIN RANGE
10 10
30
2020
30
40 40
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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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f(x) = 2X + 1
ANSWER
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Do= all real numbers
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g(x) = 1 .
X - 1
ANSWER
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Do ={ X/ X≠1}
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. .
h(x) = √ X-2
ANSWER
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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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f(x) = 2X + 1
ANSWER
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Rg = all real numbers
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g(x) = 1 .
X - 1
ANSWER
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Rg = {x/x≠0}
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. .
h(x) = √ X-2
ANSWER
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Rg={x/x≥0}