Algebra Honors (Period 6 & 7)

Functions Defined by Equations 8-7
A relation is a set of ordered pairs such as
{ (2,3), (3,5), (-4, 0), (5, 0)}
it is also a function because no repeating of the x value,


Domain of a relation is the set of 1st coordinates
the x values
Range is the 2nd coordinates
the y values

So in the example above
Domain is {2, 3, -4}
Range {3, 5, 0}

A relation that assigns to each value in the domain exactly one value in the range is called a FUNCTION
{ (2,3), (3,5), (-4, 0), (5, 0)} is a FUNCTION, whereas,

{ (2,3), (2,5), (-4, 0), (5, 0)} is NOT a function

values of domains ( x's) each are paired with only one element in the range.
Several ways to check... we looked at T- tables to compare the x values, we looked at mapping and we looked at graphs. Notice the vertical line test.

WE then compared the notation for functions
first we looked at y = 3x + 4 vs f(x) = 3x + 4

solve for x = 5
For y = 3x + 5
y = 3(5) + 4
y= 15 + 4
y = 19

Now, what was x again
Oh yeah... x = 5
so the ordered pair is (5, 19)
With f(x) = 3x + 4 however we have
f(5) = 3(5) + 4
f(5) - 15 + 4
f(5) = 19
and you can see what x was originally
(5, 19) is the ordered pair

a
Two ways to show functions
f(x) ... and we noted tht it could be g(x) or h(t) etc
or
f:x-->

g:x-->4 + 3x - x²
if the domain D = { -1, 0, 1, 2}
g:-1--> g(-1) = 4 + 3(-1) - (-1)²= 0
g:0-->g(0) = 4 +3(0) -0 = 4
g:1--> g(1) 4 + 3(1) - (1)² = 6
g:2-->4 + 3(2) - (2)²

Range = {0. 4. 6} Notice that we list each number once ( even if there is a repeated number)

f:x --> x² - 2x for the set of all REAL numbers
find
f(4) = (4)² -2(4) = 8
f(-3) = (-3) ² -2(-3) = 15
f(2) = (2)² -2(2) = 0

so the Range is {8, 15, 0}
Relations & Functions
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and the y values are the RANGE.

FUNCTIONS: Relations where there is just one y value for each x value IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
In the real world, there are excellent examples....pizza prices.
A restaurant can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.

VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function, but a vertical line function, x = 4 is not.

Any line, y = mx + b, is a function.

INPUTS: x values
OUTPUTS: y values

f(x) means the value of the function at the given x value
You can think of f(x) as the y value

Finding the value of a function: Plug it in, plug it in!
f(x) = 2x + 7
Find f(3)
f(3) = 2(3) + 7 = 13
The function notation gives you more information than using y
If I tell you y = 13 you have no idea what the x value was at that point
But if I tell you f(3) = 13, you know the entire coordinate (3, 13)

Domain of a function = all possible x values (inputs) that keep the solution real
Range of a function = all possible y values (outputs) that result from the domain

EXAMPLE:
f(x) = x + 10 has the domain of all real numbers and the same range because every value will keep the answer f(x) a real number

EXAMPLE:
f(x) = x² has the domain again of all real numbers, BUT the range is greater than or = to zero
because when a number is squared it will never be negative! So f(x) will always be 0 or positive

EXAMPLE:
f(x) = absolute value of x has the domain of all real numbers, but again the range will be greater than or equal to zero because absolute value will never be negative

EXAMPLE:
f(x) = 1/x has a domain of all real numbers EXCEPT FOR ZERO because it would be undefined if zero was in the denominator. The range is all real numbers except zero as well.
This function will approach both axes but never intersect with them.
The axes are called asymptotes which means that they will get very close but never reach them

EXAMPLE:
f(x) = (x - 10)/x + 3

Domain is all real numbers EXCEPT -3 because -3 will turn the denominator into zero (undefined)
What is the range?