Chapter 2 Functions and Linear Functions

§ 2.1 Introduction to Functions

Relation 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 #3 Section 2.1

Functions 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 #4 Section 2.1

Basic Functions Determine whether the following is a function. EXAMPLE Determine whether the following is a function. 1 drums guitar 2 violin 3 flute 4 Domain Range SOLUTION Yes, because none of the members of the domain correspond to more than one member of the range. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.1

Basic Functions Determine whether the following is a function. EXAMPLE Determine whether the following is a function. 3 ants beetles 8 crickets 5 moths 9 Domain Range SOLUTION No, because one of the members of the domain, 9, corresponds to more than one member of the range. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.1

Example: Determine whether each relation is a function: Functions 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 #7 Section 2.1

Functions as Equations 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 #8 Section 2.1

The output value, the y value, is often denoted by f(x), read Function Notation 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 #9 Section 2.1

Basic Functions Find the indicated function value: Replace x with 3 EXAMPLE Find the indicated function value: SOLUTION Replace x with 3 Evaluate the exponent Multiply Add and Subtract Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.1

Basic Functions Find the indicated function value: Replace z with -4 EXAMPLE Find the indicated function value: SOLUTION Replace z with -4 Evaluate the exponents Multiply Add Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.1

Basic Functions Find the indicated function value: Replace w with x+y EXAMPLE Find the indicated function value: SOLUTION Replace w with x+y Rewrite exponent Multiply Add Distribute NOTE: THIS CANNOT BE SIMPLIFIED ANY FURTHER!!! Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.1