Functions and Graph of Functions

Functions and Graph of Functions - Transcript

3 1 FUNCTIONS and 3 2 GRAPHS OF FUNCTIONS
Relation
A HYPERLINK http mathwords com s set htm set of HYPERLINK http mathwords com c coordinates htm ordered pairs
For example 1 2 3 4 1 a 5 r is a relation
So is the set x y y x2 this is the set of all ordered pairs x y for which y x2


Interval Notation
A notation for representing an HYPERLINK http www mathwords com i interval htm interval as a pair of numbers
The numbers are the endpoints of the interval
HYPERLINK http www mathwords com p parentheses htm Parentheses and or HYPERLINK http www mathwords com b brackets htm brackets are used to show whether the endpoints are excluded or included
For example 3 8 is the interval of HYPERLINK http www mathwords com r real numbers htm real numbers between 3 and 8 including 3 and excluding 8


Domain
The HYPERLINK http www mathwords com s set htm set of values of the HYPERLINK http www mathwords com i independent variable htm independent variable s for which a HYPERLINK http www mathwords com f function htm function or HYPERLINK http www mathwords com r relation htm relation is defined
Typically this is the set of x values that give rise to real y values

Range
The set of y values of a HYPERLINK http www mathwords com f function htm function or HYPERLINK http www mathwords com r relation htm relation
More generally the range is the HYPERLINK http www mathwords com s set htm set of values assumed by a function or relation over all permitted values of the HYPERLINK http www mathwords com i independent variable htm independent variable s


Function
A HYPERLINK http mathwords com r relation htm relation for which each HYPERLINK http mathwords com e element of a set htm element of the HYPERLINK http mathwords com d domain htm domain corresponds to exactly one element of the HYPERLINK http mathwords com r range htm range
is a function because each number x in the domain has only one possible HYPERLINK http mathwords com s square root htm square root
is not a function because there are two possible values for any positive value of x

Vertical Line Test
A test use to determine if a HYPERLINK http mathwords com f function htm relation is a function A relation is a function if there are no HYPERLINK http mathwords com v vertical htm vertical HYPERLINK http mathwords com l line htm lines that intersect the HYPERLINK http mathwords com g graph of an equation or inequality htm graph at more than one HYPERLINK http mathwords com p point htm point

Function Notation
You ve been playing with y for some time now And you ve seen that the nice equations are the ones that you can solve for y These y equations are HYPERLINK http www purplemath com modules fcns htm t self functions But the question you are facing at the moment is Why do I need this function notation and how does it work
y and f x pronounced as eff of eks mean the exact same thing but f x gives you more flexibility and more information
You used to say y 2x 3 solve for y when x 1 Now you say f x 2x 3 find f 1 pronounced as f of x is 2x plus three find f of negative one You do exactly the same thing in either case you plug in 1 for x multiply by 2 and then add the three to get a value of 1
But now you have more flexibility Your graphing calculator will list the functions as y1 y2 etc More standardly we use names like f x g x h x s t etc The point is that you can use more than one function at a time and not mix them up wondering Okay which y is this anyway Copyright Elizabeth Stapel06 2008 All Rights Reserved
How do we go about evaluating functions First remember this While parentheses have up until now always indicated multiplication look back at that circumference formula in the last paragraph above parentheses do not indicate multiplication in function notation f x means plug in a value for x it does not mean multiply f and x
Note When you have function notation the x in f x is called the argument of the function or just the argument So if they give you f 2 and ask for the argument the answer is just 2 Now that we have that straight let s proceed to evaluation You evaluate f x just as you would evaluate y
Given f x x2 2x 1 find f 2
f 2 2 2 2 2 1
4 4 1
7
Given f x x2 2x 1 find f 3
f 3 3 2 2 3 1
9 6 1
2
If you experience difficulties with HYPERLINK http www purplemath com modules negative htm negatives try using parentheses as we just did it helps keep track of things like whether or not the exponent is on the negative Remember that x is just a box waiting for something to be put into it Don t let odd looking problems scare you
Evaluate f

You can evaluate functions at variables or expressions rather than at numbers also
Given that f x 3x2 2x find f h
Everywhere that your formula has an x you now plug in an h
f x 3x2 2x
f 3 2 2
f h 3 h 2 2 h 3h2 2h
Given that f x 3x2 2x find f x h
Everywhere that your formula has an x you now plug in an x h
f x 3x2 2x
f 3 2 2
f x h 3 x h 2 2 x h
3 x2 2xh h2 2x 2h
3x2 6xh 3h2 2x 2h
If you re not sure how we got the stuff inside the parentheses the set that the 3 was multiplied through then you ll need to review HYPERLINK http www purplemath com modules polymult htm polynomial multiplication
Given that f x 3x2 2x find f x h f x h
This is actually something you will see again in calculus I guess they re trying to prep you when they give you exercises like this but it s not like anybody remembers these by the time they get to calculus so it s really a lot of work for no real purpose However this type of problem is quite popular so you should expect to need to know how to do it
The best course of action that I ve found is to break this up into pieces That is find f x h and f x and then subtract Once you ve simplified only then do you do the division It looks like this
f x 3x2 2x
f 3 2 2
f x h 3 x h 2 2 x h
3x2 6xh 3h2 2x 2h
f x 3x2 2x
f x h f x
3x2 6xh 3h2 2x 2h 3x2 2x
3x2 6xh 3h2 2x 2h 3x2 2x
3x2 3x2 6xh 3h2 2x 2x 2h
6xh 3h2 2h
f x h f x h 6xh 3h2 2h h
h 6x 3h 2 h
6x 3h 2
Be careful and split the problems into small simple steps and you should be able to complete these successfully
Piecewise functions
A HYPERLINK http www mathwords com f function htm function that uses different HYPERLINK http www mathwords com f formula htm formulas for different parts of its HYPERLINK http www mathwords com d domain htm domain

Since HYPERLINK http www purplemath com modules fcnnot htm l piecewise t self piecewise functions are defined in pieces then we have to graph them in pieces too For instance suppose we have
Since this has two pieces we will do two T charts if it had more pieces we would do more T charts The break is at x 1 so that is where our T charts will break The procedure looks like this




Why did we list that last point in T chart 1 in parentheses Because technically speaking x 1 does not belong on that chart But it is often helpful to know where the function almost is at the end That s why on the graph we drew that point as an open circle meaning that the graph is everything up to but not including that point This can be especially important when as in this case the pieces of the function don t join up at the ends