Download 2 – 1 Functions and Their Graphs

2 – 1:
Functions and Their Graphs
Objective: Ca 24: Students solve
problems involving functional concepts
such as composition, and performing
arithmetic operations on functions.
A relation is a mapping or
pairing of input values with
output values.
The set of input values is the
domain. (x – values)
The set of output values is
the range. (y – values)
A relation is a function if and
only if there is exactly one
output value for each input
value.
Example 1: Identify the domain
and range. Then tell if the
relation is a function.
INPUT
-3
1
1
4
OUTPUT
3
-2
1
-4
The domain is {-3, 1, 4}.
The range is {3, -2, -1, 4}.
This relation is not a function
because the input 1 has two
outputs –2 and 1.
EXAMPLE 2:
INPUT
OUTPUT
-1
3
1
1
3
1
4
-4
Domain = {-1, 1, 3, 4}.
Range = {3, 1, -2}.
Vertical Line Test
A relation is a function if and
only if no vertical line
intersects the graph of the
relation at more than one point.
Example 3: Determine if the
graph of a relation is a function.
2
-5
5
-2
Yes: a function
2
-5
5
-2
Yes
Graphing and Evaluating
Functions
Many functions can be
represented by an equation in
two variables such as
y = 2x + 7
An ordered pair (x, y) is a
solution of such an equation if
the equation is true when the
values of x and y are
substituted into the equation.
In an equation the input
variable is called the
independent (x) variable.
The output variable is called
the dependent (y) variable.
Graphing Equations in
Two Variables
To graph an equation in two
variables follow these steps.
1. Construct a table of values.
2. Graph enough points to
recognize a pattern.
3. Connect the points with a
smooth curve.
Example 4:
Graph the function y = x + 1
Step 1: Construct a table of
values
x
0
1
2
y=x+1
1
2
3
Step 2: Plot the points
4
2
-5
5
-2
Step 3: Draw a line through
the points
4
y = x+1
2
-5
5
-2
Linear Functions
The function y = x + 1 is a linear
function because it is of the
form y = mx + b
Where m and b are constants.
The graph of any linear function
is a line. By naming a function f
you can write the function using
function notation.
The symbol f(x) is read as “the
value of f at “x” or simply “f of x”.
f is another name for y.
f  x  x 1  y
The domain of a function consists
of all values of x for which the
function is defined.
The range of a function consists of
all values of f(x) where x is in the
domain of ‘f’.
Functions do not have to be
named ‘f’ other letters may
used, such as ‘g’, and ‘h’.
Example 5: Evaluating
Functions.
Decide whether the function is
linear. Then evaluate the
function when x = -2
f  x    x  3x  5
2
Is not a linear equation
because it has an x2 term.
Evaluating:
f  x    x  3x  5
2
f  2     2   3  2   5
2
   4   6  5
 4  6  5
7
g  x   2x  6
Is linear because it is a first degree
polynomial and is of the form
y = mx +b
g  2  2  2  6
 4  6
2
HOMEWORK