1. 1.2 Functions and Graphs
Determine whether a correspondence or a relation is a function.
Find function values, or outputs, using a formula or a graph.
Graph functions.
Determine whether a graph is that of a function.
Find the domain and the range of a function
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2. Relation
A relation is a correspondence between the first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to at least one member of the range.
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3. Function Definitions
A function is a rule or relation that assigns to each element of one set exactly one element of a second set.
The set of elements in the first set is called the domain. If x is any element in the domain, then it is called the independent variable.
The set of elements in the second set is called the range. If y is an output of the function from an input x, then y is called the dependent variable.
Functions may be defined as a set of ordered pairs, a table, a graph, an equation or a verbal statement.

4. Recognizing Functions
For ordered pairs, if the independent variable values are all different, the table is a function.
For equations, if the power on the dependent variable(y) is odd, then the equations defines the dependent variable(y) as a function of the independent variable(x).
In the case of graphs, the Vertical Line Test is the way to determine if the graph represents a function.
Vertical Line Test: a graph is a function if no vertical line intersects the graph in more than one point.
Is the table a function?
Does the equation
define y as a function of x?
x Year
N Worldwide PCs(millions)
1991
129.4
1992
150.8
1993
177.4
1994
208
1995
245
2000
535.6
2005
903.9

5. Which graphs represent a function?b)

6. Example (continued) No. Yes. Yes.
Which of graphs (d) - (f) (in red) are graphs of functions? In graph (f), the solid dot shows that (–1, 1) belongs to the graph. The open circle shows that (–1, –2) does not belong to the graph.
No.
Yes.
Yes.
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7. Function Notation
We can use the function notation , read “y equals f of x” to indicate that the variable y is a function of the variable x. For specific values of x, f(x) represents the resulting y values especially when referring to a graph generated by the function. The point (a, f(a)) lies on the graph for any given a in the domain of the function. For example: represents a function and for

8. Example {(9, 5), (9, 5), (2, 4)} {(–2, 5), (5, 7), (0, 1), (4, –2)}
Determine whether each of the following relations is a function. Identify the domain and range.
{(9, 5), (9, 5), (2, 4)}
{(–2, 5), (5, 7), (0, 1), (4, –2)}
{(–5, 3), (0, 3), (6, 3)}
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9. Example – Evaluating a Function
A function is given by f(x) = 2x2  x + 3. Find each of the following.
a. f (0) b. f (–7) c. f (5a) d. f (a – 4)
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10. Example – Using the Graph
For the function f (x) = x2 – 6, use the graph to find each of the following function values.
a. f (‒3) b. f (1)
Locate the input ‒3 on the horizontal axis, move vertically (up) to the graph of the function, then move horizontally to find the output on the vertical axis.
f (‒3) = 3
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11. Domain of a Function Example: Find the domain. a) b) c) d)
To make sure that the function results in a real number value, watch for two types of exceptions:
Values from the domain that make the denominator 0 (zero).
Values in the domain that make a negative value under an even root.
Example: Find the domain. a) b) c) d)

12. Example Graph f (x) = x3 – x . What is the domain of this function?
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13. Example (continued) Graph What is the domain of the function?
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14. Example Domain = (–, ) Range = [‒4, )
Graph the function. Then estimate the domain and range.
Domain = (–, )
Range = [‒4, )
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