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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 2.1 Functions and Their Representations

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Objectives Basic Concepts Representations of a Function Definition of a Function Identifying a Function Tables, Graphs and Calculators (Optional)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. The notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f. Name y = f(x) Output Input FUNCTION NOTATION

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. The variable y is called the dependent variable and the variable x is called the independent variable. The expression f(4) = 28 is read f of 4 equals 28 and indicates that f outputs 28 when the input is 4. A function computes exactly one output for each valid input. The letters f, g, and h, are often used to denote names of functions.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Representations of a Function Verbal Representation (Words) Numerical Representation (Table of values) Symbolic Representation (Formula) Graphical Representation (Graph) Diagrammatic Representation (Diagram)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Evaluate f(x) at the given value of x. f(x) = 5x – 3 x = 4 Solution f(4) = 5(4) – 3 = 20 – 3 = 23

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3). Solution Verbal Representation Multiply a purchase of x dollars by 0.06 to obtain a sales tax of y dollars. Numerical Representation xf(x)f(x) $1.00$0.06 $2.00$0.12 $3.00$0.18 $4.00$0.24

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3). Solution Symbolic Representation f(x) = 0.06x Graphical Representation

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3). Solution Diagrammatic Representation

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Let function f square the input x and then add 3 to obtain the output y. a. Write a formula for f. b. Make a table of values for f. Use x = 2, 1, 0, 1, 2. c. Sketch a graph of f. Solution a. Formula If we square x and then add 3, we obtain x Thus the formula is f(x) = x

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) b. Make a table of values for f. Use x = 2, 1, 0, 1, 2. c. Sketch the graph. Solution xf(x)f(x)

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair: (x, y) InputOutput Definition of a Function A relation is a set of ordered pairs, and a function is a special type of relation.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value. Function The domain of f is the set of all x-values, and the range of f is the set of all y-values.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Use the graph to find the functions domain and range. Domain Range

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Use the graph to find the functions domain and range. Domain Range

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Use f(x) to find the domain of f. a. f(x) = 3xb. Solution a. Because we can multiply a real number x by 3, f(x) = 3x is defined for all real numbers. Thus the domain of f includes all real numbers. b. Because we cannot divide by 0, the input x = 4 is not valid. The domain of f includes all real numbers except 4, or x 4.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Slide 17

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Determine whether the table represents a function. xf(x)f(x) The table does not represent a function because the input x = 3 produces two outputs; 4 and 1.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. If every vertical line intersects a graph at no more than one point, then the graph represents a function. Vertical Line Test

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Determine whether the graphs shown represent functions. a.b. Passes the vertical line test. The graph is a function. Does not pass the vertical line test. The graph is NOT a function.

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