2.2 FunctionsA. Define functions

B. Function notationC. Evaluate Functions

D. DomainE. Difference Quotient

A. Different ways relations can look:

• {(4,2),(2,1),(3,2),(1,4)} is a relation.• y = x + 1 is a relation.• This graph is a relation:

• A RELATION just relates #s to #s….x’s to y’s

A FUNCTION, however, is

• a relation in which each input has a UNIQUE (only one) output.

• i.e., a relation in which each x-value has only one y-value partnered with it.

{(4,2), (2,1), (3,2), (1,4)} is a function

• (input 4 and input 3 have the same output, but that’s OK) (y’s don’t have to be faithful.)

{(4,2), (3,1), (4,0)} is NOT a function(input 4 has more than only one output) (x’s are

not allowed to cheat on their partners!)

Vertical Line Test

• If you can draw/imagine a vertical line anywhere on the graph and the line touches more than 1 spot, then it FAILS and is NOT a function. Some examples:

B. “f(x)” is read “f of x”

• f(x) is NOT f times x• “f(x)” is the same thing as “y” except that it

denotes that what we are dealing with is indeed a function.

C. If f(x) = 3 – 2x, then find f(-1)

• Means plug in (-1) wherever an x is.

Try: Find f(-1/2) for f(x) = 3 - 2x

Let g(x)=-x2+4x+1. Find g(x+2).

• Skeleton: -( )2 + 4( ) + 1

Try: h(t) = t2 – 2t. Find h(x + 2).

A piecewise function looks like:

x2 + 1, if x < 0f(x) = x – 1, if x ≥ 0

Imagine the conveyor belt…

Which machine does it go into? Depends on what the input is.

For that last piecewise f’n,

• Find f(-1).

• Find f(0).

• Find f(2).

Try:

4 – 5x, if x ≤ -2• f(x) = 0, if -2 < x < 2

x2 + 1, if x > 2Find f(-3).Find f(4).Find f(-1).

Here are a few problems where the instructions are important.

• “Find all real values of x such that f(x) = 0.”• f(x) = 5x + 1• First replace f(x) with zero.

• Then solve for x.

You try: “Find all the real values of x such that f(x) = 0.”

• f(x) = x2 – 8x + 15

Another set of interesting instructions:

• Find the values of x for which f(x) = g(x) if f(x)=x4-2x2 and g(x)=2x2.

• First set f(x) = g(x).

• Solve for zero and solve.

You try: Find all values of x for which f(x)=g(x) if f(x)=x2+2x+1 and g(x)=3x+3.

D. Find the domain:

3( )

3 5

yf x

y

( ) 2 3f x x

E. This is called a difference quotient:

0,)()(

hh

xfhxf

It’s a special formula you will derive in your calculus class. You will not have to memorize this formula for your algebra exams. It will be provided to you if we want you to use it on the test.

Why are we looking at it before you are in calculus?

Getting from this formula to an answer will be all algebra-esque, lots of distributing, combining like terms, etc.

Find and simplify the difference quotient if f(x) = x2 – x + 1

The formula for the difference quotient (which will be given to you) is

Let’s do this one piece at a time. First let’s find f(x+h) = (_____)2 – (_____) + 1Then let’s find f(x) = (that was an easy one)Then put it all together…

0,)()(

hh

xfhxf

This time, x will be replaced with the number 2. f(x) = x2 – x + 1

• Find the difference quotient: 0,)2()2(

hh

fhf