3.1 Functions and their Graphs Relation – a mapping, or pairing of input values with output...
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Transcript of 3.1 Functions and their Graphs Relation – a mapping, or pairing of input values with output...
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3.1 Functions and their Graphs
Relation – a mapping, or pairing of input values with output values.
Domain – set of input valuesRange – set of output values
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Functions
Function – a relation is a function if there is exactly one output for each input.
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Relations and Functions
Relations and functions between 2 quantities can be represented in many ways:
-mapping diagrams-tables-graphs-equations-verbal descriptions
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Functions and Relations
Relations can be represented by ordered pairs (x, y) where x-coordinate is the 1st number and y-coordinate is the 2nd number.
Domain = First number (input)Range = Second number (output)
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Functions and Relations
Quadrant IQuadrant II
Quadrant IVQuadrant III
y-axis
x-axis
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Relations and Functions
Consider the following points:
{(a, 1), (b, 2), (c, 3), (e, 2)}List the domain: {a, b, c, e}List the range: {1, 2, 3}
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Relations and Functions
Ex 2: {(3, 5), (4, -6), (2, -4), (-1, 5)}
List the Domain:{-1, 2, 3, 4}
List the Range: {-6, -4, 5}
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Relations and Functions
How to tell if a relation is a function:-Only one output for each input (no x can be repeated)-Vertical Line Test: = no vertical line intersects the graph of the relation at more than 1 point.
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Relations and Function
For the relation to be a function, no x may be repeated
Are the following Functions?1. {(1, 3), (-4, 2), (-6, 2), (0, 5)}
Yes = no x has been repeated2. {(1, 3), (-4, 2), (-6, 7), (1, 5)}
No = 1 was repeated
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Relations and FunctionsInput Output
Age Weight16 220 Write as Ordered Pairs
17 125 (16, 220), (16, 122), (17, 179)
18 179 (18, 125), (18, 116)
116 Not a function!!
122
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Relations and FunctionsInput OutputName Weight
Sue 125Mary 133Steve 159Carol 144Jose
Write as ordered pairs:{(Sue, 125), (Mary, 133), (Steve, 159), (Carol, 144), (Jose, 133)}
Yes, it is a function – no Input has been repeated
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Relations and Functions
Vertical Line Test1. Is this a Function? Yes
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Vertical Line Test, cont
Are the following functions?
No
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Relations and Functions
Many functions can be represented by an equation in 2 variables:
Ex: y = 2x – 7 An ordered pair (x, y) of the equation is
a solution of the equation if the ordered pair is true when the values of x and y are substituted into it.
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Relations and Functions
Ex: for the line y = 2x – 7, is the ordered pair (2, -3) a solution?
Substitute the values in for x and y
-3 = 2 (2) – 7-3 = -3 YES, the ordered pair is a
solution of the equation.
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Relations and Functions
Are the following solutions to the equations?
1. y = 3x – 1 ; ((2, 5), (-2, -7)yes, yes
2. 2p + q = 5; (2, 3) (-5, 15)no, yes
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Relations and Functions
In an equation, the input variable (x, domain) is the independent variable, and the output (y, range) is the dependent variable because it depends on the value of the input.
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3.3 Functions - ContinuedFunction Notation – the symbol f(x) is read “f of x” and is used to notation a function.
Since a function is a relation, a function can be listed as a set of ordered pairs
(x, f(x)) where the domain is all values for which
the function is defined, and the range consists of the values of f(x) where x is the domain of f.
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Functions, Cont
To determine the range of a given function (given the domain), simply plug the values in for the variable.Ex: f(x) = 3x + 2 Domain: {-1, 0, 5}
f(-1) = 3(-1) + 2 = -1f(0) = 3(0) + 2 = 2f(5) = 3(5) + 2 = 17
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Functions, Cont
Find the range of f(x) = 2x – 7
given the D {-3, -1, 0, 7}
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Functions, cont
Find the Domain of x:We assume the domain of a function to
be all real numbers that are an acceptable replacement for the variable (x).
To find the domain of a function, we must determine whether there are any unacceptable replacements.
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Unacceptable Replacements
2 Things that make unacceptable replacements:
1. 0 in the denominator – if a value would make the denominator = 0, then the value is unacceptable.
2. (-) under the radical – if a value would cause the expression under the radical to be a negative number, then the value would be unacceptable.
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Domain of a FUNCTION
Find the domain of:f(x) =
What happens if x = -3?f(-3) = = Undefined
Therefore the domain of the function is
D = {x| x 3}Which reads all x such that x does not equal -3
3
4
x
x
0
7
33
43
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Domain of a function
Find the domain of the following functions:
1. 5.
2. 6.
3. 7.
4. 8.
x
xxf
2
23)(
72)( xxf
)6(
7)(
x
xxf
)2)(5()(
xx
xxf
xxf )(
8)( xxf
23)( xxf
43
4)(
x
xxf
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Homework
p. 109 (13-20)
p. 114 (9, 11, 19)
p. 119 (1-27 odd)