warm up 1. find a triple if r = 10 and s = 2. 2. if a = 6, b = 8 and the triangle is a right...
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Warm Up
1. Find a triple if r = 10 and s = 2.2. If a = 6, b = 8 and the triangle is a
right triangle, what is the value of c? 3. If c = 50 and b = 48, what is the value of a if it is a right
triangle? 1
32
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Lesson 4 Functions and Function Notation
Objective: To learn about functions, domains and ranges. To use function notation to represent functions
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Functions
y = 3x + 1
For this equation, every x we choose, will give us a new y.
x is the input, and y is the output. For every x we choose there is one
and only one y possible – therefore this is a FUNCTION
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Definition of a Function
A function is a correspondence between two sets X and Y that assigns to each element x of set X exactly one element y of set Y
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Domain and Range
For each element x in X, the corresponding element y in Y is called the image of the function at x. The set X is called the domain of the function, and the set of all function values, Y is called the range of the function.
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Ex 1: Determine whether each relation is a function.
a. {(4,5), (6,7), (8,8)}
b. {(5,6), (4,7), (6,6), (6,7)} We begin by making a figure
for each relation that shows set , the
domain, and set , t
So
he
luti
ra
o
e.
n
ng
X
Y
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Solution for part (a)
468
578
X Y
Domain Range
The figure shows that every
element in the domain
corresponds to exactly one
element in the range.No two ordered pairs in the given relation have
the same first component different second
components. Thus, the relation is a function
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Solution for part (b)
456
6
7
X Y
Domain Range
The figure shows that 6
corresponds to both 6 and 7.
If any element in the domain corresponds to
more than one element in the range, the
relation is not a function, Thus, the relation
is not a function.
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Vertical Line Test (Pencil Test)
Given a graph – any vertical line drawn on the graph should only go through one point.
FunctionNot a Function
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Function Notation
( )
The variable is called the
, because it can be assigned any
of the permissible numbers from the domain.
The variable is called the
independent
variable
dependent
, because
var its iable value depe
y f x
x
y
nds on .x
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Function Notation
The special notation f(x), read “f of x”, represents the value of the function at the number x.
The notation does not mean “ f times x.”
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Finding a Function’s Domain
The domain (the x’s) of a function can be almost any real number. There are two cases when there are exceptions: Division by zero – if x is in the bottom
of the fraction (denominator), then x cannot make that denominator 0.
f(x) = ; x = 3 is not defined. 1
3x
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Finding a Function’s Domain
Even roots of negative numbers y = is only a real number if x ≥
11x
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Finding a Function’s Domain
Exclude from a function's domain real
numbers that cause division by zero
and real numbers that result in an even
root of a negative number.
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Ex 5: Find the domain of each function
2a. ( ) 8 5 2
2b. ( )
5
c. ( ) 2
f x x x
g xx
h x x
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Solution part a
2The function ( ) 8 5 2 contains
neither division nor an even root.
The domain of is the set of all real numbers.
f x x x
f
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Solution part b
2The function ( ) contains division.
5Because division by zero is undefined, we
must exclude from the domain values of
that cause 5 to be 0. Thus cannot equal
to 5. The domain of function is
g xx
x
x x
g
{ | 5}.x x
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Solution part c
The function ( ) 2 contains an even
root. Because only non-negative numbers
have real square roots, the quantity under the
radical sign, 2 must be greater than or
equal to 0. Thus, 2 0 or 2
Th
h x x
x
x x
erefore the domain of is { | 2}
or the interval [ 2, ).
h x x
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Practice Exercises
2
Find the domain of each function:
121. ( )
361
2. ( )2
xh x
x
f xx
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Answers
1. { | 6, 6}x x x
2. { | 2} or ( 2, )x x
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Graphing Functions
Using a graphing calculator Put the equation or function into the
form y = …. Button at top left y = Then type in the rest of the equation
and press [GRAPH] at top right.
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Evaluating Functions
Given f(x) = x2 + 2x – 1, find f(2). This means to plug in 2 wherever
there is an x in the function. f(2) = (2)2 +2(2) – 1
= 4 + 4 – 1 = 7
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Evaluating Functions
Given f(x) = x2 + 2x – 1, find f(–3). f(–3) = (–3)2 +2(–3) – 1
= 9 – 6 – 1 = 2
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Practice
f(x) = x2 + 3x- 4, find f(-2) & f(0)
f(x) = x2 + 1, find f(-3) & f(x-1)
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Practice
Find the numbers for x whose image is 2. f(x) = x2