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Chapter 1: Section 1.1
Functions and Models
Section 1.1: Four Ways to Represent a Function
Definition: A Function is a rule that assigns to each input
value x exactly out-
put value y = f(x). The variable x is called the variable, and y
is called the
variable. The is the set of all allowable x values, and the
is the set of all possible y values.
An algebraic description of a function exists when there is an
explicit formula for y = f(x).
Example 1: Find the domain, the range, the value of f(3) and
sketch the graph of the following
functions:
(a) f(x) = x ...this is called the identity function
(b) f(x) = 2x 4
Example 2: Find the domain and range for each of the
following
(a) f(x) =
p2x 4 ...does f(1) exist here?
2-4>-0
t :#
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Chapter 1: Sec1.1, Four Ways to Represent a Function
(b) f(x) =
2xpx
24x ...does f(1) exist here?
If an explicit formula does not exist then we can sometimes
define a function graphically
Example 3, Reading the information from a graph: The graph of a
function f is shown in the
following figure
(a) Find the values of f(2), f(0) and f(2).(b) What are the
domain and range of f?
3 2 1 0 1 2 3
3
2
1
1
2
3
x
f(x)
Vertical Line Test: A curve in the xy-plane is the graph of a i
no vertical line
passes through the curve more than .
2 Spring 2017, Maya Johnson
XZ . 4 > 0
Yeats.
*XfIDomain:(-oo,o)U(4Tfl
1) does Not exist , since =1 not in domain .
(a) fl - 2) =-1-
/ ]flz )=
g- ( b ) Domain : 2 , 3 ]Raise :[ - I , 3 ]
functiononce
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Chapter 1: Sec1.1, Four Ways to Represent a Function
Example 3: Determine whether each is a function of x.
(a)
x
f(x)
(b)
x
f(x)
(c)
x
f(x)
3 Spring 2017, Maya Johnson
/ || / / /Fx ' is a function
|||| / fsxsmisohnyfuohetinoieisfilled in , so only one y -
valuefor that x - value .
p.p.pe.f is Not a fund . ,
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Chapter 1: Sec1.1, Four Ways to Represent a Function
Example 4 Given the graph of f(x) and g(x) below, determine the
following:
4 3 2 1 0 1 2 3 4
4
3
2
1
1
2
3
4
x
y
f(x)
g(x)
(a) f(1)
(b) g(0)
(c) The values(s) of x for which f(x) = 3
(d) The values(s) of x for which f(x) = g(x)
The four possible ways to represent a function are:
(1)
(2)
(3)
(4)
Example 5: Let f(x) = x
2+ 2x 1 and g(x) = 1
x2 , find the following
(a) f(1)
(b) g(8)
(c) f(x+ h)
4 Spring 2017, Maya Johnson
,
.
IF' EHIII
Algebraically ( with a formula )
Through words ( in a word problem )
Graphically
Numerically ( a table of data )
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Chapter 1: Sec1.1, Four Ways to Represent a Function
(d)
f(x+h)f(x)h
Definition: Functions whose definition involve more than one
rule are called Piecewise Functions.
To graph, graph each rule over the appropriate portion of the
domain.
Example 6, Absolute value function: The absolute value of a
number x, denoted by |x|, is thedistance from x to 0 on the real
number line. The plot and definition of absolute value function if
given
below.
x
y
|x| =(x x 0x x < 0
Example 7: Find the domain and sketch graph of the function
f(x) =
8 0
Domadnisl
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Chapter 1: Sec1.1, Four Ways to Represent a Function
Increasing and Decreasing Functions:
A function f is called % on an interval I if f(x1) < f(x2)
whenever x1 < x2 inI. It is called & on an interval I if
f(x1) > f(x2) whenever x1 < x2 in I.We can see from the
figure below that the function f(x) = x
2is decreasing on the interval
and increasing on the interval .
x
y
8 Spring 2017, Maya Johnson
