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INTRODUCTION TO FUNCTIONSINTRODUCTION TO FUNCTIONS
©2006 by Kelly Howarth©2006 by Kelly Howarth
What is a function? What is a function?
“A function is like an “input-output” machine. The machine RELATES the number we Input in a
certain way to the Output.
“Here’s an example of a function. Our Input numbers will be x’s and our Output
results will be y’s.
2 3y x
“Here, we see that whatever number we replace the variable x with gets doubled,
then 3 is subtracted from that product. The resulting number is y. So, if we choose an
input value (x value) of 2, what is our output value (y value) ?
“Seems to me that we could DECIDE to
choose substitute ANY NUMBER for x. How about if we see what
happens if we choose a few different numbers?”
“We can make a table for our Input values and our Output values. Let’s call
this an x-y table.
y (Outputs)y (Outputs) -4x + 2-4x + 2
(Relation)(Relation)
X (Inputs)X (Inputs)
4 2y x
“OOPS! Uh,. . .er. . .Caught us
kissing! Well, anyway, it is OBVIOUS that we
enjoy a RELATIONSHIP!”
“I enjoy several relationships since I date several girls at the same time! Heh!
Heh! ”
“YOU CAD! You, one guy, are trying to date several girls at the same time!”
Math Definition of a RelationMath Definition of a Relation
• In math, if we have one x input that has In math, if we have one x input that has MORE THAN ONE y output, then we have MORE THAN ONE y output, then we have a relation. Here’s what we mean. a relation. Here’s what we mean.
• Specifically, if we have several y’s for the Specifically, if we have several y’s for the same x, then we have a relation. Here is a same x, then we have a relation. Here is a picture next. . .picture next. . .
2x y
“We are MARRIED! She is the ONE LOVE of my life!! We have a
relationship that functions very well!!”
“He is sooo Hot! Here’s a
hot math definition for
you next!”
Definition of a Function: Definition of a Function:
• A function is a relation in so that every A function is a relation in so that every Input value (x’s) have ONLY ONE output Input value (x’s) have ONLY ONE output value (y’s). value (y’s).
• Or. . .Or. . .• For every x, there is ONLY ONE y.For every x, there is ONLY ONE y.• Here is an example of a function; Here is an example of a function;
y x
y x
“So, if he tried to have relationships with other
women after we are married, then he won’t
function!
“So, just like a marriage is ONE man with only ONE
woman; in a math function, each x has only one y. Each input number has only one output number. A function
is like marriage.”
“And just as a guy can carry on several relationships while dating (as
long as he doesn’t get caught!); Mathematically, if one x (input
number) has SEVERAL y’s (output numbers), then we have a
RELATION. A RELATION is like dating.”
Function notationFunction notation
( ) 3 8f x x 3 8y x
“So, why would I want to write f(x) instead of y if they mean the same thing? That’s
just MORE WORK!”
“Here’s why you might want to use the “function notation”
f ( x ) instead of merely writing y. Watch this
closely!”
2
2
( ) 2
( )
( ) 16 20 12
f x x
g x x
s t t t
“I can use a function to figure out how far I will
travel if I crawl at 3 miles per hour! Watch this!”
( ) 3s t t
s = distance crawled; and is our output
t = time in seconds; and is our input
Let's pick a few input values; How far will the baby have traveled
if he crawled in a straight line for 10 seconds? 2 seconds? 30 seconds?
Function examples. . .Function examples. . .
3( ) 2 Find (1)
(1) ?
( 2) ?
f x x x f
f
f
Evaluate the following functions Evaluate the following functions
2( ) 4; (4) ? ( 3) ? (0) ?f x x f f f
( ) 2 ; ( ) ? ( ) ?f x x f a b f
Assignment: Assignment: