Math 150 – Fall 2015 Section 4A 1 of 4
Definition.
• A function is a relation in which no two different ordered pairs have the same first element.
• A function can also be defined as a rule that assigns exactly one element in set
B to each element in set A .
• Set A , the set of all first coordinates, is called the domain of the function.
• Set B , the set of all second coordinates, is called the range of the function.
Graph:
Example 1.
Determine which of the following relations are functions. Also, give the relation’s domain and range.
(a) { (2 , 6) , (5 , 6) , (7 , 6) , (8 , 9) , (9 , 2) } .
(b) { ( − 2 , 5) , (3 , 4) , ( − 1 , 3) , ( − 2 , 3) , (5 , 7) }
(c) x y
− 1 − 4
0 8
1 − 10
2 5
2
3
7
− 4
(d) x y
− 1 − 5
0 − 5
1 5
2
3
5
5

Math 150 – Fall 2015 Section 4A 2 of 4
Definition.
We often use the notation f ( x ), read as “ f of x ” instead of y to give the output for the input x . That is y = f ( x ). Using f ( x ) instead of y allows us to state that f is a function, give that function a name f , and it gives us the input value x .
Example 2.
Input x Function Rule: f ( x ) = 2 x − 4 Output f ( x ) Ordered Pair ( x, f ( x ))
2
-3
0
Example 3.
Which of the following statements are true of the statement that f (3) = − 13.
1.
f · 3 = − 13 so f =
− 13
8
2. 3 is the input.
3.
f (3) is the output.
4.
− 13 is the output.
5. (3 , − 13) is an ordered pair of the function f .
Evaluation: For functions, we are given a rule, for example tells us for the input x we have the output 3 x 2 f ( x ) = 3 x 2 − 2 x + 1, that
− 2 x + 1. To evaluate the function, any input we are given is put into this rule and replaces the x .
Example 4.
For f ( x ) = 3 x 2 − 2 x + 1, find the following.
(a) f (3) =
(b) f ( − 2) =
(c) f (3 t ) =
(d) f ( a + b ) =

Math 150 – Fall 2015 Section 4A 3 of 4
Definition.
The difference quotient is the average rate of change of the y -values over the x -values in the interval [ x, x + h ], where h = 0. Graphically, it is the slope of the line that goes through the points ( x, f ( x )) and ( x + h, f ( x + h )). For h = 0, the slope of this line is m = f ( x + h ) − f ( x )
( x + h ) − x
= f ( x + h ) − f ( x ) h
Note.
You will need to MEMORIZE the difference quotient for the exam! And we
ALWAYS want to simplify our answer so that the h in the denominator can be canceled.
Example 5.
For the following functions, evaluate the difference quotient. Fully simplify your answer.
(a) f ( x ) = 3 x 2 − 2 x + 1
(b) f ( x ) =
√
2 x − 4

Math 150 – Fall 2015 Section 4A 4 of 4
√
Definition.
For a function f ( x ) defined by a rule, such as f ( x ) = the function is assumed to be any real number where f ( x ) is defined.
x , the domain of
Finding the domain: To find the domain, ask yourself what values of x would make the function undefined. The two problems we encounter the most often are:
• no denominator can be 0.
• the inside of any even root must be positive.
Example 6.
Find the domain and range of the function f ( x ) =
√ x − 7 + 3.
Example 7.
Find the domain of the following functions.
(a) f ( x ) =
√
4 x − 1 x 2 − 4 x +2
(b) f ( x ) = q
2 x − 3 x +4
(c) f ( x ) =
√
3 x 2 − 2 x +3 x 2 − 4
(d) f ( x ) =
5
√ 2 if x < 3 x if x ≥ 4