Lesson 1-1

Example 1
OCEANOGRAPHY  The table below shows the pounds per square inch (psi) of pressure that push on a body at various water depths. State the relation of the data as a set of ordered pairs. Also, state the domain and range of the relation.

Water Pressure	
Depth (ft)	Pressure (psi)	
0	14.7	
16.5	22.05	
33	29.4	
49.5	36.75	
66	44.1	
82.5	51.45	
99	58.8	
115.5	66.15	

Relation:	{(0, 14.7), (16.5, 22.05), (33, 29.4), (49.5, 36.75), (66, 44.1), (82.5, 51.45), 
(99, 58.8), (115.5, 66.15)} 
Domain:	{(0, 16.5, 33, 49.5, 66, 82.5, 99, 115.5}
Range:	{14.7, 22.05, 29.4, 36.75, 44.1, 51.45, 58.8, 66.15}

Example 2
The domain of a relation is consecutive even integers between and including -4 and 4. The range, y, of the relation is three more than half x, where x is a member of the domain.  Write the relation as a table of values and as an equation. Then graph the relation. 

Table:	x	y		Graph:		
		-4	1				
		-2	2				
		0	3				
		2	4				
		4	5				

						
Equation:  

Example 3
State the domain and range of each relation.

a.		b.		
It appears from the graph that all real numbers are included in the domain and the range.	It appears from the graph that all real numbers are included in the Domain. The range includes all negative numbers and zero.	

Example 4
State the domain and range of each relation. Then state whether the relation is a function. 

a.	{(-1, 2), (0, 4), (1, 3)} 
The domain is {-1, 0, 1} and the range is {2, 3, 4}. Each element of the domain is paired with exactly one element of the range, so this relation is a function. 

b.	{(-3, 1), (-2, 5), (-3, 4), (1, 2)} 
The domain is {-3, -2, -1} and the range is {1, 2, 4, 5}. In the domain, -3 is paired with two elements of the range, 1 and 4. Therefore, the relation is not a function. 

Example 5
Determine if the graph of each relation represents a function. Explain. 

a.		b.		
No, the graph does not represent a function. Values of x > 0 are paired with two elements of the range. Therefore, it fails the vertical line test.	Every element of the domain is paired with exactly one element of the range. Thus, the graph represents a function.	

Example 6
Evaluate each function for the given value. 

a.	f(-2) if f(x) = 2x4 - 3x3 + 4x - 1	b.	g(3) if g(x) = |5 - x3|
	f(-2)	= 2(-2)4 - 3(-2)3 + 4(-2) - 1	g(3)	= |5 - (3)3|
	= 32 + 24 - 8 - 1	= |5 - 27|
	= 47	= |-22|
	= 22

Example 7
Evaluate each function for the given value.

a.	h(b) if h(x) = 3x4 - 2x3 + x2 - x
h(x)	= 3(b)4 - 2(b)3 + (b)2 - b	x = b
	= 3b4 - 2b3 + b2 - b

b.	j(2a - 1) if j(x) = 2x2 - 5x + 1
j(2a - 1)	= 2(2a - 1)2 - 5(2a - 1) + 1	x = 2a - 1
	= 2(4a2 - 4a + 1) - 5(2a - 1) + 1
	= 8a2 - 8a + 2 - 10a + 5 + 1
	= 8a2 - 18a + 8

Example 8
State the domain of each function. 

a.	f(x) = 	b.	g(x) = 	
Any value that makes the denominator equal zero must be excluded from the domain of f since division by zero is undefined. To determine the excluded values, let x2 - 6x + 8 = 0 and solve.
	x2 - 6x + 8 = 0
	(x - 2)(x - 4) = 0
	x = 2 or x = 4
Therefore, the domain includes all real numbers except 2 and 4.	Any value that makes the radical negative must be excluded from the domain of g since the square root of a negative number is not a real number. Also, the denominator cannot be zero. Let x + 3  0 and solve for the excluded values.
	x + 3	 0
	x	 - 3
The domain excludes numbers less than or equal to -3. The domain is written as
{x | x > -3}, which is read “the set of all x such that x is greater than -3.”