Domain/RangeContinuous/Discontinuous
Increasing/DecreasingConstant
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Objectives• I can find domain and range in Interval
Notation• I can identify increasing, decreasing, and
constant intervals of a function• I can tell if a function is continuous
Domain and Range
• The domain in any relation is the first coordinates from the ordered pairs. It is the Input!
• Domain = X -Values• The range in any relation is the second
coordinates from the ordered pairs. It is the Output!
• Range = Y- Values
Example 1: Domain/Range
• Given the following relation• {(2,3), (-4,8), (2,6), (7,-3)}• What is the Domain?• { -4, 2, 7}• **Notice they are listed least to greatest!! • No duplicates!!!• What is the Range?• {-3, 3, 6, 8}
x
y
4
-4
The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists.
The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain.
Domain
Range
x
y
– 1
1
Example: Find the domain and range of the function f (x) = from its graph.
The domain is [–3,∞).
The range is [0,∞).
3x
Range
Domain
(–3, 0)
Example 1Domain( , )
Range[ 3, )
Example 2
Domain( , )
Range( , 4]
Example 3
Domain[0, )
Range( , )
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Functions
INCREASING
DECREASING
CONSTANT
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• decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2),• constant on an interval if, for any x1 and x2 in the interval, f (x1) = f (x2).
The graph of y = f (x):
• increases on (– ∞, –3),
• decreases on (–3, 3),
• increases on (3, ∞).
A function f is:• increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) < f (x2),
(3, – 4)
x
y(–3, 6)
–2
2
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2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
7
123456
8
-2-3-4-5-6-7
Look at the graph of the function shown on the interval (-6,-2)
This means x values between –6 and –2.
As you follow the graph of the function from x = -6 to x = -2, does the function value (remember that is the y value) increase, decrease, or remain constant (the same)?
It INCREASES so we say the function is increasing on the interval (-6, -2)
Can you see another interval where the function is increasing?
The function is also increasing on (4, 6)
x = 4x = -6 x = -2 x = 6
This is NOT an ordered
pair
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2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
7
123456
8
-2-3-4-5-6-7
Can you see an interval where the function is decreasing?
The function is decreasing on the interval (-2, 4) since when you follow the graph between x = -2 and x = 4 the function value (y value) goes down.
Remember for an interval you list the x values that make the y values decrease. Always move from left to right on the graph (from smaller x values to larger x values). x = 4x = -2
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2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
7
123456
8
-2-3-4-5-6-7
What is this function doing on the interval (-7, -2)?
It is
INCR
EASI
NGx = -2x = -7
What is this function doing on the interval (-2, 2)?
What is this function doing on the interval (2, 7)?
x = 2 x = 7
It is DECREASING
It is not increasing OR decreasing but remaining
constant
Continuous or Discontinuous??
• A function is continuous if it has an infinite domain and forms a smooth line or curve
• Simply put: It has NO BREAKS!!!
• You should be able to trace it with your pencil from left to right without picking up your pencil
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Look at the following graphs and determine if they are Continuous
or Discontinuous Functions??
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Homework
• WS 1-3• Quiz next class• Work on Parent Function Packet