1.2 FUNCTIONS
You should learn to:
1. Decide whether a relation between two variables is a function.
2. Find domains and ranges of functions .
3. Use function notation and evaluate functions.
4. Use functions to model and solve real-life problems.
Terms to know: relation, function, domain, range, independent variable, dependent variable,
--
f\lnction notation,
'
domain
~s;defu;ed {o;Piecewise ) flJilction, implied domain versus specified
A relation is ~ set of ordered pairs.
A function f from a set A to a set B is a relation that assigns to each element (member) in the
~
set A exactly one element in the set B.
(A function has no repeating x-values and passes the vertical line test)
~
G}
Set A is the domain (or set of inputs) and Set B is the range (OI"set of outputs) for the function
f
(Domain: x-values and Range: y-values)
,~.e~,.,..,.~...A OoW~QM
e;~
.. Al
1
ck~rveVlr,L
1
Inputs
ou>t-puk
\Jq, ft410
\.)6-Example 1: Decide whether or not each relation is a function. For each function, list the domain
and range.
'Z"'
e
b. {(0, 2),(1, 5), ~ ,(~,42}
'\' ~+-
a.
SetA
b.
a.
Input, x
Output, y
42
2
1
2
2
2
3
f?~'trv--)
Functions are usually represented by equations. For example, y =2x + 1 represents the variable
y as a function of the variable x. x is the independent variable and y is the dependent
variable. (The value of y depends upon the chosen value for x). The domain is the set of all x
values, and the range is the set of ally values.
'11f1i
Example 3: Which of the equations represents y as a function
a. x
2
+ y =1
r
'a-::
'2..
-X
+-
1,
'"t UY\.GT '
-
x =1
•
Ji j
(fY'\
l
c. y+lxl=1
'0-= 1-/xl
~·
M
/
.
equation of a function. For example, y
--E
No+
j\)o
J
(x), g ( x), and ~ to represent y
= 2x + 1 could be written as
in the
f ( x) = 2x + 1 in function
notation.
f ( x)
Note:
does not mean " f times x ."
Example4: Let f(x) = -x2 +2x+1, g(x) = ~ , h(x) = 2x 2
,
and k(x) = l.
Evaluate (fmd) the following:
a. /(3)
b.
-0),.-+ z_(__i:) + 1
g(3)
V-.3
-L.
f(-3)
f.
-(-~2 +-L(-3)+ l
- q
-~
+
f(a)
'2.
- ()\
+~o.
-l-J
g. h(-3)
2(-~z..
~
_@
j. g(2x-1)
V3-z (zx-1)..
'3 - liK t- 2.
k(3)
@
-J ~-L{-~
131.
. - /Lj
i.
g(-3)
d.
d. (~')1.
·Vi -"2..(3)
-G) +-l=, + I
e.
c. h(3)
k. h(x +1)
'Q
h. k(-3)
I. k(x
c)l~~tx~0
2-( ~z.i-b: ~ {
~·
I
f. x=b
~ ~i;y, , ,.
Function notation uses expressions such a.:_f
-
- v '2.
-Efj-
~fund-~
'X
\..1=-:t~
u + . r:::-
x~ 9
d.
y=mx+b
'2..
~ =- 1-
2
+ 1)
1
Js-- L(x.
43
A piecewise-defmed (piecewise) function is a function that is defmed by two or more equations
overa
iii-
::~:~:mf(x)=~
-~:~-~~
l:-,.
find:
X> I
•
/(--:2)
b. /(-1)
;)_- (_-7::)
(= 1)1...
BJ
[[]
a.
f.
Graph the piecewise function
c.
/(0)
d. /(1)
·o~~[§]
~~m
r"'
~
~
~!l
f .
~
~~
~
~
,...
X
I
'i~ _{'
-1
0
1
!
f
I
0
I
f '-l
Example 6: What is the domain for each of the functions below?
2
a. f(x)=-x +2x+1
b.
g(x)= e
3-Z..x :2 0
.1
c. h(x)=~
~'2.-i -:::_0
~ ~-J~+L =- 0
~
)(\
~
d.
I
Domain Restrictions: 1. Fractions
2. Even Roots
Qc&L ~ cct N:i
D
2
,•
Sometimes, it is easier to find the domain and range of a function by looking at its graph.
.
44
~
m
e. /(2)
~ ()o le.~ t-o ( .'ejrJ
0 off\0.. \\"
() _
bo~
o
to t-op
~~~
Example 7: Find the domain and range from graphs:
a.
f(x)=-8
y
1
b. g(x) C )
' Y
X
OcVWl.~ ~
J
E:""'1 -~ ug) ~J ... ,.
,
I
\.
J
X'~O
~
~
~
~Q~b
c-~)o]
1\1"'11111....
I
._
.,
~
....
~
,,... ~
x~4Cio
~'t._ l{ =- .o
ea
c:c-- ~(~+1)=0
X~ 2. X = - L
.
'Ill
- fJ."" ")
~~ .. ~·.2
oo~ i Y' ~c-eO) o) u (o) ~:1
~ ~- ':--
K~ ~
i)
~
0 (-~ ) ())
u ~) 010)
0
((
Example 8: A company invests $42,000 for equipment to produce a product. Each
unit of the product cost.s $2.30 and is sold for $5.98. Let xbe the number of units
produced and sold.
A. Write the total cost Cas a function of
B. Write the revenue R as a function of x.
C. Write the profit Pas a function of x.
' {Note: P = R - C) . :
D :. 3. ·~ ~ ;. - 42.00b
Example 9: For
a.')
t~e,function f(x)
B ind
~(;j:: 0
sef et()Lachan ~ 0
sbl~ ~ x
~)( -S" =- 0
~ Jl- ::.
~
•
[~ =
s
x.
( C?) ::: ~ ·'S 0
+ 1.{21 000
X.
R(~ := S. q % X
p (x') ==-
.s-. 'l1f x s~q2
\f:>&\ ~
1:,
X-
. ~%
G.3 oP
"'"l.-30~
lf2
fib
-\.('l.O
'1-. - Y 200D]
all real values ofx such that a) f(x)=O and b) f(O)
(p)~co)
+cO)=- 3(0;• -s-
/!l~= -5]_
s-;3]
45