Download 2.3 Functions Definition

2.3 Functions
Definition: A relation is any set of ordered pairs; such as
4, 1 ,
2,3 , 0, 5 , 2,7 , 3, 9
If the value of the second coordinate, y, depends on the value of the first coordinate, x, then y is
the dependent variable and x is the independent variable. An example of a relation is the amount
paid for gas at a gas station depends on the number of gallons dispensed. This could be
described by the following set; 5, $12.75 , 8, $20.40 , 13. $33.15 , 20, $51 , 25, $63.75 .
Definition: A function is a relation in which no two different ordered pairs have the same first
coordinate.
Another example of relation is the amount of horsepower a Ford Lightning produces depends on
the revolutions per minute (rpm) that the engine is turning. This could be described by the
following set of ordered pairs;
(rpm,horsepower) = 4000,330 , 4500,360 , 4750.370 , 5000,375 , 5250,370 , 5500,360 . Does
this relation describe a function? The answer is yes, since no two different ordered pairs have the
same first coordinate.
Definition: The domain of a relation is the set of all first coordinates and the range is the set of
all second coordinates.
Examples: Determine if the relation defines a function. State the domain and range.
1.
2, 8 , 1, 1 , 0,0 , 1,1 , 2,8
2.
2,4 , 1,1 , 0,0 , 1,1 , 2,4
3.
4.
4,8 ,
2,3 ,
1,0 , 1,1 ,
2,8 , 3, 2
5.
Solution: 1 is a function; domain 2, 1,0,1,2 and range 8, 1,0,1,8
2 is a function; domain 2, 1,0,1,2 and range 0,1,4
3 is not a function, since there are two ordered pairs namely 2,3 and
the same first coordinate; domain 4, 2, 1,1,3 range 8,3,0,1, 2
4. is a function; domain 4,6,7, 3, 2 and range 100,200,300,400
5. is not a function, since there are two ordered pairs namely 1,6 and
the same first coordinate; domain 3, 1,1,0,2 range 8,6,4,2,0, 4
2,8 with
1,2 with
6. Graph the relation
and determine if y defines a function of x.
Solution: Make a table of values
4
2
1
1
0 0
1 1
4 2
The answer is no, since there are two ordered pairs with the same x-coordinate namely 4,2 and
4, 2 .
Vertical Line Test: The graph of a relation is the graph of a function if and only if every vertical
intersects the graph in at most one point.
From the preceding example we see that a vertical line intersects the graph at two points,
therefore the equation does not define y as a function of x.
In example 7-9, decide whether each relation defines a function.
7
8.
9.
Solution: Examples 8 and 9 are functions (pass the vertical line test), but example 7 is not.
Recall: The domain of a function is the set of all first coordinates (x-coords) and the range is the
set of all second coordinates (y-coords).
For the following examples, state the domain and range of each function.
Increasing, Decreasing, and Constant Functions: A function increases on an interval of its
domain if its graph rises from left to right on the interval. It decreases on an interval if its graph
falls from left to right on the interval. It is constant on an interval of its domain if its graph is
horizontal on the interval.
For the following examples, determine the intervals of the domain for which each function is
(a) increasing, (b) decreasing, and (c) constant.
The domain of a function f is the set of all real numbers x such that
is a real number.
In the remaining examples, determine the domain of each function.
20.
Solution; Since f is a rational function, its domain is the set of all real numbers except those
which make the denominator equal to zero.
Domain: |
6
21.
Solution; Since f is a rational function, its domain is the set of all real numbers except those
which make the denominator equal to zero.
2
7
2
7
8
12
6
2
The domain is |
6,2
22.
5
√
Solution; Since f is a square root function, its domain is the set of all real numbers except those
which make the radicand negative (since the square root of a negative number is an imaginary
number)
Domain:
5 0
5
The domain of the function is 5, ∞
23.
5
4
Solution: The function tells us to square a number, subtract five times the number, and then add
4. Since these operations can be performed on any real number, we conclude that the domain of f
is all real numbers. The Domain is ∞, ∞ .