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**Functions and Their Representations**

Section 2.1 Functions and Their Representations

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**Objectives Basic Concepts Representations of a Function**

Definition of a Function Identifying a Function Tables, Graphs and Calculators (Optional)

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FUNCTION NOTATION 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

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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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**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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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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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 x f(x) $1.00 $0.06 $2.00 $0.12 $3.00 $0.18 $4.00 $0.24

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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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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 1 ● 2 ● 3 ● 4 ● ● 0.06 ● 0.12 ● 0.18 ● 0.24

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Example Let function f square the input x and then add 3 to obtain the output y. Write a formula for f. Make a table of values for f. Use x = −2, −1, 0, 1, 2. 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) = x2 + 3.

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Example (cont) b. Make a table of values for f. Use x = −2, −1, 0, 1, 2. c. Sketch the graph. Solution x f(x) −2 7 −1 4 3 1 2

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**Definition of a Function**

A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair: (x, y) Input Output A relation is a set of ordered pairs, and a function is a special type of relation.

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Function A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value. 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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**Example Use the graph to find the function’s domain and range. Range**

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**Example Use the graph to find the function’s domain and range. Range**

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Example Use f(x) to find the domain of f. a. f(x) = 3x b. 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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**Example Determine whether the table represents a function.**

f(x) 2 −6 3 4 −1 1 The table does not represent a function because the input x = 3 produces two outputs; 4 and −1.

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Vertical Line Test If every vertical line intersects a graph at no more than one point, then the graph represents a function.

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**Example Determine whether the graphs shown represent functions. a. b.**

Does not pass the vertical line test. The graph is NOT a function. Passes the vertical line test. The graph is a function.

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