Functions and Their Representations Section 2.1 Functions and Their Representations

Objectives Basic Concepts Representations of a Function Definition of a Function Identifying a Function Tables, Graphs and Calculators (Optional)

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

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.

Representations of a Function Verbal Representation (Words) Numerical Representation (Table of values) Symbolic Representation (Formula) Graphical Representation (Graph) Diagrammatic Representation (Diagram)

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

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

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

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

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 x2 + 3. Thus the formula is f(x) = x2 + 3.

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

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.

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.

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

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

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.

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

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.