Domain/Range Continuous/Discontinuous Increasing/Decreasing Constant

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 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 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,∞). Range Domain Example: Domain & Range (–3, 0)

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 decreasing on an interval if, for any x 1 and x 2 in the interval, x 1 f (x 2 ), constant on an interval if, for any x 1 and x 2 in the interval, f (x 1 ) = f (x 2 ). The graph of y = f (x): increases on ( – ∞, – 3), decreases on ( – 3, 3), increases on (3, ∞). Increasing, Decreasing, and Constant Functions A function f is: increasing on an interval if, for any x 1 and x 2 in the interval, x 1 < x 2 implies f (x 1 ) < f (x 2 ), (3, – 4) x y ( – 3, 6) –2–2 2

Copyright © by Houghton Mifflin Company, Inc. All rights reserved 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 = 4 x = -6x = -2 x = 6 This is NOT an ordered pair

Copyright © by Houghton Mifflin Company, Inc. All rights reserved 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 = 4 x = -2

Copyright © by Houghton Mifflin Company, Inc. All rights reserved What is this function doing on the interval (-7, -2)? It is INCREASING x = -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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

Look at the following graphs and determine if they are Continuous or Discontinuous Functions?? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

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