1.2 Functions

Pre-Calculus

Introduction to Functions

Many everyday phenomena involve pairs of quantities that are related to each other by some rule of correspondence. The mathematical term for such a rule of correspondence is a relation.

1. The simple interest I earned on an investment of $1000 for 1 year is related to the annual interest rate r by the formula I=1000r.

2. The area A of a circle is related to its radius r by the formula 2A r

Function

Not all relations have simple math formulas. People commonly match up NFL starting quarterbacks with touchdown passes, and the hours of day with temperature. In each of theses cases, there is some relation that matches each item from one set with exactly one item from a different set.

This is called a function.

Definition of a Function

Function – from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B.

Domain – Set A (input) Range – Set B (output)

Characteristics of a Function from Set A to Set B

1. Each element of A must be matched with an element of B.

2. Some elements of B may not be matched with any element of A.

3. Two or more elements of A may be matched with same element of B.

4. An element of A (the domain) cannot be matched with two different elements of B.

Testing for Functions

Example 1: Decide whether the table represents y as a functionof x.

X Y

-3 5

-1 -12

0 5

2 3

4 1

Testing for Functions Represented Algebraically

Which of the equations represent(s) y as a function of x?

(a)

(b)

2 1x y

2 1x y

Function Notation

When an equation is used to represent a function, it is convenient to name the function so that is can be referenced easily. For example, you know that the equation y = 1 - x2 describes y as a function of x. Suppose you give this function the name “f”.

Input (x) Output f(x) Equation f(x) = 1 – x2

Evaluating a Function

Let g(x) = -x2 + 4x +1. (a) g(2)(b) g(t)(c) g(x + 2)

A Piecewise-Defined Function

Evaluate the function when x = -1 and x =0.

2 1, 0( )

1, 0x x

f xx x

Domain of a Function

The domain of a function can be described explicitly or it can be implied by the expression used to define the function. The implied domain is the set of all real numbers for which the expression is defined.

Examples

(a)

(b)

2

1( )4

f xx

( )f x x

Finding the Domain of a Function

Find the domain of each function.

(a) f: {(-3, 0), (-1, 4), (0, 2), (2, 2), (4, -1)}

(b) g(x) = -3x2 +4x +5(c) h(x) = 1/(x+5)(d) Volume of a sphere: (e)

343

V r( ) 4 3k x x

Cellular Communications Employees

The number N (in thousands) of employees in the cellular communications industry in the U.S. increased in a linear pattern from 1998 to 2001. In 2002, the number dropped, then continued to increase through 2004 in a different linear pattern. These two patterns can be approximated by the function

Where t represents the year, with t=8 corresponding to 1998. Use this function to approximate the number of employees for each year from 1998 to 2004.

23.5 53.6,8 11( )

16.8 10.4,12 14t t

N tt t

Baseball

A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and an angle of 45 degrees. The path of the baseball is given by the function

f(x) = -0.0032x2 + x +3 where x and f(x) are measured in feet. Will the baseball clear a 10-foot fence located 300 feet from home plate?

Homework

Page 24-292-10 even, 13-23 odd, 28-42 even, 53-61 odd,

79-82 all, 85