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: