One quantity depends on another quantity # of shirts -> revenue Age -> Height Time of day -> Temperature

3.1 Function

Function-Definition

4 Representations of a function:• Verbally – • Numerically – • Algebraically – • Visually –

Formal definition: A function f is a rule that

assigns to each element x in a set A exactly

one element, f(x), in a set B. ( f(x) is read as “ f of x ”)

Verbal & Numeric

Verbal: Function f(x) divides x by 3 and then adds 4.

A B

Numeric:

x -2 5 9 -4 1

f(x)

Table describes the values in A and their assignments in B.

Algebraic & Graphic

Algebraic:

Graphic:

x -2 5 9 -4 1

f(x) 41 42 -3 27 5

Plot points (x, f(x)) in the two dimensional plane

43

)( x

xf

2 4 6 8

-21

21

42

x

f(x)

Example: Evaluate the function at the indicated values.

3)( 2 xxf

xfxffff ,1),2(),2(),0(

Piecewise Functions

0 if

03- if

3 if

4

5

3

)(2 x

x

x

xx

x

xg

Evaluate the function at the indicated values.

)2(),0(),1(),3(),4( ggggg

)4(g

)3(g

)1(g

)0(g

)2(g

Difference QuotientFind f(a), f(a+h), and the difference quotient for the given function.

7)( 2 xxf

Difference quotient:h

afhaf )()(

Domain

Domain: the set of all inputs for which the function is a real number. (valid x)

Recall definition: A function f is a rule that assigns to each

element x in a set A exactly one element, f(x), in a set B.

Set A = input => x = independent variable

Set B = output => y = f(x) = dependent variable

Two restrictions on the domain for any function:

Example 1

Find the domain for the following function.

36

5)(

2

x

xxf

D:

Example 2Find the domain for the following function.

xxf 11)(

D: