Presentation on theme: "Functions and Their Representations"— Presentation transcript:
1 Functions and Their Representations Section 2.1Functions and Their Representations
2 Objectives Basic Concepts Representations of a Function Definition of a FunctionIdentifying a FunctionTables, Graphs and Calculators (Optional)
3 FUNCTION NOTATIONThe notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f.Namey = f(x)Output Input
4 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.
5 Representations of a Function Verbal Representation (Words) Numerical Representation (Table of values) Symbolic Representation (Formula) Graphical Representation (Graph) Diagrammatic Representation (Diagram)
6 ExampleEvaluate f(x) at the given value of x. f(x) = 5x – 3 x = −4 Solution f(−4) = 5(−4) – 3 = −20 – 3 = −23
7 ExampleLet 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 Representationxf(x)$1.00$0.06$2.00$0.12$3.00$0.18$4.00$0.24
8 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
9 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 Representation1 ●2 ●3 ●4 ●● 0.06● 0.12● 0.18● 0.24
10 ExampleLet 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.Solutiona. Formula If we square x and then add 3, weobtain x Thus the formula is f(x) = x2 + 3.
11 Example (cont)b. Make a table of values for f. Use x = −2, −1, 0, 1, 2. c. Sketch the graph. Solutionxf(x)−27−14312
12 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 OutputA relation is a set of ordered pairs, and a function is a special type of relation.
13 FunctionA 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.
14 Example Use the graph to find the function’s domain and range. Range
15 Example Use the graph to find the function’s domain and range. Range
16 ExampleUse 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.