Under Function Basics, we saw the difference between a relation and a function.
In either case, we are dealing with relationships expressed as ordered pairs.


functionfactory1

All of the values that can go into a relation or function (input) are called the domain.

All of the values that come out of a relation or function (output) are called the range.
Range may also be referred to as "image".

Note that both relations and functions have domains and ranges.

The domain is the set of all first elements of ordered pairs (x-coordinates).
The
range is the set of all second elements of ordered pairs (y-coordinates).
Only the elements "used" by the relation or function constitute the range.

definition
Domain: all x-values that are to be used (independent values).
Range: all y-values that are used (dependent values).

Set Builder notation may be used to express domains and ranges.

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ex2

State the domain and range of the following relation:
{(1,3), (-2,7), (3,-3), (4,5), (1,-3)}.

State whether the relation is a function.

Solution: Domain: {-2, 1, 3, 4}.     Range: {-3, 3, 5, 7}.
While these listings appear in ascending order, ordering is not required.
Do not, however, duplicate an element.


No, this relation is not a function. The x-value of "1" had two corresponding y-values (3 and -3).

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ex1

eyecolorSmall
State the domain and range of the following relation:
(eye color, student's name)
A = {(blue, Steve), (green, Elaine), (brown, Kyle), (blue, Marsha),
         (brown, Miranda), (green, Dylan)}

State whether the relation is a function.

Solution: Domain: {blue, green, brown}.    Range: {Steve, Elaine, Marsha, Miranda, Dylan}.
No, this relation is not a function. The eye colors are repeated with different students.

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ex3

State the domain and range for the elements matched in the diagram below.
State whether the matches form a function.

domainpic

Solution: Domain: {3, 4, 5, 6, 7}.     Range: {1, 2, 9, 12}.
Note that the range is only the elements that were used.

Yes, the relation {(3,2), (4,1), (5,9), (6,12), (7,12)}is a function.
No x-value has more than one y-value.

FYI: Set B = {1, 2, 3, 8, 9, 12} may be called the co-domain. It is the "possible" set from which output from the relation will fall. The co-domain is NOT necessarily the same as the range. There may be values in the co-domain that are never used.

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ex4

State the domain and range associated with the scatter plot shown below.
State whether the scatter plot is a function.

domaingraph1

Solution: Domain: {1, 2, 3, 4, 5, 6}.
(Be careful not to simply list the domain as 1 < x < 6, which would imply ALL values (including fractions) between 1 and 6 inclusive, unless you specify "x is an integer".)     

Range: { 0, -1, 1, 2, 3, 6}

Yes, this is a function. No x-values have more than one associated y-value, and it passes the Vertical Line Test for functions.

Note: Graphs that are composed of a series of dots, instead of a connected curve, are referred to as discrete graphs. A discrete domain is a set of input values that consist of only certain numbers in an interval.

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ex5

State the domain and range associated with the graph below.
State whether this relation is a function.

domaingraphpic2

Solution: Domain: domain22 (all real numbers). 
The arrows indicate that the graph continues off the visible grid, so assume that all real numbers are involved.    

Range: y > -3   (may also be written as setnotation5a)

Yes, this relation is a function, since it passes the Vertical Line Test for functions.

Note: Graphs that are composed of a connected curve are referred to as continuous graphs. A continuous domain is a set of input values that consists of all numbers in an interval.

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ex6

State the appropriate domains for the functions shown below.

When functions are expressed as "rules" (formulas), be sure to think about possible "problem" spots before stating the domain. Remember that the domain is the x-values that are allowed by the function's equation. In most cases, a graph will help show the domain.

a) y = x + 6

Solution: This is the equation of a straight line. There are no "problem" spots with this straight line.
Domain: All real numbers, domain22

domaingraph7

b) domainmath9

Solution: If a fractional expression contains a variable in its denominator, you need to check for division by zero. Set x² - x - 6 = 0 to find the problem spots of x = -2 and x = 3 which cause a zero denominator in this example..
Domain: domain22 - {-2} - {3}. 

domaingraph8

c) domainmath10

Solution: The value under the radical needs to be 0 or a positive number (no negatives). To find the values that are OK, set x + 2 > 0, which is x > -2.
Domain:   x > -2.

domaingraph9


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statement
It may be necessary to restrict a domain to ensure the existence of a function.

We know not all graphs are functions. As we saw in Example 6, it is often possible, however, to create "functions" from non-function graphs by restricting which domain elements are used.

domaingraphpic2a
The graph at the left is domainmath5.

At first glance, it appears that this graph passes the Vertical Line Test and is a "function". But it is NOT a function over the domain of Real numbers.
This graph is undefined when x = 2, because substituting two into the function will create a zero denominator.    

If we restrict the domain to be "all Real numbers excluding 2", our relation can be called a function.

Domain of the function: domainmath6        


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statement
It may be necessary to restrict the range to ensure the existence of a function.

The graph of the relation relationmath1 is shown below on the left. It is clear that this relation fails the Vertical Line Test and is NOT a function. We can, however, separate this graph into its two parts and create two separate functions.
domainmath7
domaingraphpic3a
pic5
math5
pic6
math6
These separated graphs each pass the Vertical Line Test and are functions. The domain for both functions is x > 0. The range of the first function is y > 0, and the second function is y < 0.

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hint gal
If a domain is not stated, it is generally assumed to be all real numbers.


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