Algebra II Analyzing and Working with Functions Part 1 2015-04-21 - PDF document

Slide 1 / 166 Slide 2 / 166 Algebra II Analyzing and Working with Functions Part 1 2015-04-21 www.njctl.org Slide 3 / 166 Table of Contents click on the topic to go Part 1 to that section Function Basics Operations with Functions Composite Functions The 12 Basic Functions (Parent Functions) Part 2 Transforming Functions Inverse Functions Piecewise Functions

Slide 4 / 166 Function Basics In this section, we will review functions and relations, function notation, domain, range, along with discrete and continuous functions. These topics were also covered in 8th grade and Algebra 1. Return to Table of Contents Slide 5 / 166 Relations A relation is an association between sets of information. Slide 6 / 166

Slide 7 / 166 Graphs of Functions The Vertical Line Test can determine if a graph represents a function. If the vertical line intersects only one point at a time on the ENTIRE graph, then it represents a function. If the vertical line intersects more than one point at ANY time on the graph, then it is NOT a function. Move the black vertical line to test! Slide 7 (Answer) / 166 Graphs of Functions The Vertical Line Test can determine if a graph represents a function. If the vertical line intersects only one point at a time on the ENTIRE Function Not a Function graph, then it represents a function. If the vertical line intersects more than one point at ANY time on the graph, then it is NOT a function. Teacher Notes Move the black vertical line to test! [This object is a pull tab] Slide 8 / 166 Equations as Functions An equation is a function only if a number substituted in for x produces only 1 output or y -value. Function Reason Not a Function Reason y = 3 x + 4 x = 5 For each input There are multiple for x, there is only values for y. one output of y. y = 5 All y values x = y 2 For each x, there are 5. are two values for y.

Slide 9 / 166 Slide 9 (Answer) / 166 Slide 10 / 166 Function Determine if each of the relations below is a function and provide an explanation to support your answer: x y x y x y -2 2 7 5 -1 3 4 3 -3 8 2 8 -5 4 9

Slide 10 (Answer) / 166 Function Determine if each of the relations below is a function and provide an explanation to support your answer: x y x y x y -2 2 7 5 -1 3 3 -3 8 4 Not a 2 8 -5 4 9 Function - Function Function Answer x value of - each x - each x -1 yields value has value has more a unique a unique than one y value y value y value [This object is a pull tab] Slide 11 / 166 Function Determine if each of the relations below is a function and provide an explanation to support your answer: Slide 11 (Answer) / 166 Function Determine if each of the relations below is a function and provide an explanation to support your answer: Not a Function - x value of 2 yields two different y values; does not pass vertical line test. Answer Not a Function - multiple x values produce more than one y value; does not pass vertical line test. Function - No x values repeat. [This object is a pull tab]

Slide 12 / 166 Slide 12 (Answer) / 166 Slide 13 / 166 1 Is the following relation a function? {(3,1), (2,-1), (1,1)} Yes No

Slide 13 (Answer) / 166 1 Is the following relation a function? {(3,1), (2,-1), (1,1)} Yes Answer Yes No [This object is a pull tab] Slide 14 / 166 2 Is the following relation a function? X Y -2 3 Answer Yes 0 2 -1 -1 No 3 2 -2 0 Slide 15 / 166 3 Is the following relation a function? Yes No

Slide 15 (Answer) / 166 3 Is the following relation a function? Yes Answer No Yes [This object is a pull tab] Slide 16 / 166 Slide 16 (Answer) / 166

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Slide 19 / 166 Function Notation So why change the notation? 1) It lets the mathematician know the relation is a function. 2) If a second function is used, such as g( x ) = 4 x, the reader will be able to distinguish between the different functions. 3) The notation makes evaluating at a value of x easier to read. Slide 20 / 166 Evaluating a Function To Evaluate in y = Form: To Evaluate in Function Notation: Find the value of y = 2 x + 1 Given f(x) = 2x + 1 find f(3) when x = 3 y = 2 x + 1 f(3) = 2(3) + 1 y = 2(3) + 1 f(3) = 7 y = 7 "f of 3 is 7" When x is 3, y = 7 Similar methods are used to solve but function notation makes asking and answering questions more concise. Slide 21 / 166

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Slide 23 / 166 Slide 23 (Answer) / 166 Slide 24 / 166 6 Given and Find the value of .

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Slide 26 / 166 8 Given and Find the value of . Slide 26 (Answer) / 166 8 Given and Find the value of . Answer [This object is a pull tab] Slide 27 / 166

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Slide 30 (Answer) / 166 Slide 31 / 166 Slide 32 / 166 Interval Notation Inequality Notation Graph a b Closed Interval a b Open Interval a b Half-Open Interval

Slide 33 / 166 Slide 34 / 166 Summary { } = set D = Domain (possible input or x-values) R = Range (possible output or y-values) ∈ = is an element of (belongs to) = positive infinity = negative infinity = Set of Real Numbers = Set of Integers = Natural Numbers Slide 35 / 166 Infinity Why do you think parentheses are used in interval notation for a data set that includes or instead of brackets?

Slide 35 (Answer) / 166 Infinity Why do you think parentheses are used in interval notation for Infinity/Negative Infinity a data set that includes or instead of brackets? do not have a final value, Answer they can always increase/ decrease. Since they never end a parentheses is used instead of a bracket. [This object is a pull tab] Slide 36 / 166 13 What is the interval notation for the given graph? A B C D E F Slide 36 (Answer) / 166 13 What is the interval notation for the given graph? A B C Answer A D E F [This object is a pull tab]

Slide 37 / 166 14 What is the inequality notation for the given graph? A B C D E F Slide 37 (Answer) / 166 14 What is the inequality notation for the given graph? A B Answer C E D E F [This object is a pull tab] Slide 38 / 166 15 What is the interval notation for the given graph? A B C D E F

Slide 38 (Answer) / 166 15 What is the interval notation for the given graph? A B Answer C B D E F [This object is a pull tab] Slide 39 / 166 16 What is the inequality notation for the given graph? A B C D E F Slide 39 (Answer) / 166 16 What is the inequality notation for the given graph? A B C Answer D F E F [This object is a pull tab]

Slide 40 / 166 Domain and Range The domain of a function or a relation is the set of all possible input values (x-values). The range of a function or a relation is the set of all possible output values (y-values). Slide 41 / 166 Domain and Range Relation Domain Range Slide 42 / 166 Domain and Range State the domain and range for each example below: x y 2 7 3 -3 8 4 x y -2 3 4 -5 x y 5 1 8 2 9

Slide 42 (Answer) / 166 Domain and Range State the domain and range for each example below: Domain Range x y 2 7 3 -3 Answer 8 4 x y -2 3 4 -5 [This object is a pull tab] x y 5 1 8 2 9 Slide 43 / 166 Domain and Range State the domain and range for the function below. Write your answers in inequality and interval notation. Slide 43 (Answer) / 166 Domain and Range State the domain and range for the function below. Write your answers in inequality and interval notation. Answer Domain: -2 ≤ x < 2 and [-2, 2) Range: -2 ≤ y < 4 and [-2, 4) [This object is a pull tab]

Slide 44 / 166 Domain and Range State the domain and range for the function below. Write your answers in inequality and interval notation. Slide 44 (Answer) / 166 Domain and Range State the domain and range for the function below. Write your answers in inequality and interval notation. Answer Domain: Range: [This object is a pull tab] Slide 45 / 166

Slide 45 (Answer) / 166 Slide 46 / 166 18 Is -2 < x < 2 the domain of the relation? Yes No Slide 46 (Answer) / 166 18 Is -2 < x < 2 the domain of the relation? Yes Answer No No Domain: -2 ≤ x ≤ 2 [This object is a pull tab]

Slide 47 / 166 19 Is [0, 1] the range of the relation? Yes No Slide 47 (Answer) / 166 19 Is [0, 1] the range of the relation? Yes Answer No Yes [This object is a pull tab] Slide 48 / 166

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