§ 2.1 Introduction to Functions and sets from 1.1

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 1.1 Sets p 6 The braces, { }, indicate that we are representing a set. This form of representation, called the roster method, uses commas to separate the elements of the set. The ellipsis indicates that there is no final element and that the listing goes on forever. The objects in a set are called the elements of the set. Such as: A set is a collection of objects whose contents can be clearly determined.

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 1.1 Set-Builder Notation p 6 {x | x is a real number and greater than 10} Express x > 10 using set-builder notation EXAMPLE SOLUTION The set of all x Such that X is a real number greater than 10

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 1.1 Symbols p 7 and The symbol is used to indicate that a number or object is in a particular set. Here is an example: 7 {1,2,5,7,9} The symbol is used to indicate that a number or object is not in a particular set. For example: 3 {4,6}

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.1 Relation p 96 Definition of a Relation A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation.

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.1 Relation p 96 Check Point 1 Find the domain and the range of the relation: {(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (34, 15.5)} Domain: {0, 10, 20, 30, 34} Range: {9.1, 6.7, 10.7, 13.2, 15.5} What is the rule that assigned the “inputs” in the domain to the “outputs” in the range? For example, for the ordered pair (30, 13.2), how does the data in Figure 2.1(a), which shows the percentage of first-year US college students claiming no religious affiliation Domain Range

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.1 Functions p 98 Definition of a Function A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range. (Each x corresponds to exactly one y in a function. Values of x have to be “faithful” – so to speak – that is, each x goes to exactly one y. However, a y value may be the image of more than one x – so y’s are not required to be “faithful”) For example, the following relation is a function: {(1,2), (2,3), (3,5), (4,3)} Note that the y value 3 is the image of two x values.

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.1 Basic Functions 98EXAMPLE SOLUTION Determine whether the following is a function guitar violin drums flute Yes, because none of the members of the domain correspond to more than one member of the range. DomainRange

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.1 Basic Functions 98EXAMPLE SOLUTION Determine whether the following is a function beetles crickets ants moths No, because one of the members of the domain, 9, corresponds to more than one member of the range. DomainRange

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.1 Functions 99 Example: Determine whether each relation is a function: (a) {(2,3), (4,5), (6,5), (9,10)} (a)Yes – a function (b) {(1,2), (3,3), (6,8), (1,10)} (b) No – not a function since 1 is mapped to 2 and 10 (c) {(1,5), (4,5), (2,5), (3,5)} (c) Yes – a function

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.1 Relation p 99 Check Point 2 Determine whether each relation is a function: a. {(1,2), (3, 4), (5, 6), (5, 7)} b. {(1, 2), (3, 4), (6, 5), (7, 5)} Not a function function

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.1 Functions as Equations 99 Functions are usually given in terms of equations rather than as sets of ordered pairs. For example, consider the function y = 2x – 3 For each value of x, there is one and only one value of y. The variable x is called the independent variable and the variable y is called the dependent variable.

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.1 Function Notation 100 If an equation in x and y gives only one value of y for each value of x, then the variable y is a function of the variable x. When an equation represents a function, the function is often named by a letter such as f, g, h, F, G, or H. The output value, the y value, is often denoted by f(x), read “f of x” or “f at x” The notation f(x) does not mean “f times x”. The notation describes the value of the function f at x.

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.1 Basic Functions 100EXAMPLE SOLUTION Find the indicated function value: Replace x with 3 Evaluate the exponent Multiply Add and Subtract

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.1 Basic Functions 101EXAMPLE SOLUTION Find the indicated function value: Replace z with -4 Evaluate the exponents Multiply Add

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 2.1 Basic Functions 101 Check Point 3a. SOLUTION Find the indicated function value: Replace x with 6 Multiply Add and Subtract

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 2.1 Basic Functions 100 Check Point 3b. SOLUTION Find the indicated function value: Replace x with -5 Evaluate the exponent Multiply Add and Subtract

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 2.1 Basic Functions 100 Check Point 3c. SOLUTION Find the indicated function value: Replace r with -4 Evaluate the exponent Multiply Add and Subtract

Blitzer, Intermediate Algebra, 5e – Slide #19 Section 2.1 Basic Functions 101 Check Point 3d. SOLUTION Find the indicated function value: Replace x with a+h Multiply NOTE: THIS CANNOT BE SIMPLIFIED ANY FURTHER!!!

DONE

Blitzer, Intermediate Algebra, 5e – Slide #21 Section 2.1 Basic Functions 101EXAMPLE SOLUTION Find the indicated function value: Replace w with x+y Rewrite exponent Multiply Add Distribute NOTE: THIS CANNOT BE SIMPLIFIED ANY FURTHER!!!