A Function is ... (3.1)

A Function is ... (3.1)

                                 
A DIFFERENCE QUOTIENT EXAMPLE:

A FUNCTION is...
a "RULE" that explains what mathematical operations will be APPLIED to:
- an INPUT (an "X") to produce the OUTPUT (the "Y" value)
- "X" (the INDEPENDENT VARIABLE)
- a NUMBER from the DOMAIN
-  to get the VALUE for Y (the DEPENDENT VARIABLE) from a given "X" VALUE

the DOMAIN is the set of possible "X" values
the RANGE is the set of  "Y" values
use letters like "f" or "g" or "h" ... to represent a function

f can also be thought of as a MAPPING (Arrow diagram) 
showing how (CONNECTS) each
value from the DOMAIN is
matched to a PARTNER in the RANGE

f can also be pictured as a MACHINE that takes
a given INPUT and produce a SINGLE OUTPUT
using a UNIQUE RULE
"x" is an INPUT and
"y" or "f(x)" is the OUTPUT value

f(x) is read as
- f of x
- f operating on x
- f evaluated at x
- the value of f at x
- the image of x under f

An employee for ACME INDUSTRY earns $9 for each hour worked.
To calculate the EARNINGS for a certain individual we
"Multiply the NUMBER OF HOURS (x) by 9"
g(x) = 9 times x
g(x) = 9x
suppose the DOMAIN is {20,30,40}
then the RANGE will be {180, 270, 360}
g(20) = 9(20) = 180
etc.

If ACME INDUSTRY changes the pay rate to $360 for any
employee that works less than or equal to 40 hours in a given week.
When an employee works more than 40 hours they receive an
additional $10 for each hour over 40. This creates the need for a
special type of function called a PIECEWISE DEFINED FUNCTION.
Two different rules are needed. One rule for less than or equal to 40 hours
and a second rule for over 40 hours.

g(x) = 360  .....................  if   0< x <= 40
g(x) = 360 + 10(x-40)  .... if 40 < x

REPRESENTING A FUNCTION:
1) VERBALLY (using words)
2) ALGEBRAICALLY (using a formula ... f(x) = 9x   )
3) VISUALLY (use a GRAPH)
4) NUMERICALLY (by use of TABLE)