MA112 – 2.2
Today:
2.1 homework word problems
2.2: the algebra of functions, finding
domains
Announcements
 Homework 2.2 due Mon
 quiz on finding domain
Section 2.2 Algebra of Functions (+,-,×, ÷)
I.
Importance of the function domain
dictionary definitions for domain:
- place of residence if there is absolute ownership / rule
- field of expertise or influence
The domain of a function is the set of all values where the function works; i.e., where the
function is able to give an answer.
Example: f ( x) 
1
x
For most values of x, f (x) has an answer:
1 1
x = 1  f ( x)    1
x 1
1 1
x = 2  f ( x)    0.5
x 2
1
 0.1
x = -10  f ( x) 
 10
However, when x=0, f (x) fails to give an answer:
1
x = 0  f ( x)   ???
0
We say that x=0 is outside the function’s domain.
With humor, we could say that f (x) doesn’t have The Power to give the output for f(0).
However this specially defined function P (x ) does have The Power:
1
 x
P( x)   17
1
 x
for x  0
for x  0
for x  0
II.
Arithmetic Combinations
Functions can work together to make new functions.
Consider two functions f (x) and g (x ) .
Notation
Sum:
Difference:
Product:
Quotient:
f (x ) + g (x )
f (x ) - g (x )
f (x ) * g (x )
f (x ) / g (x )
 f  g (x)
 f  g (x)
 f  g (x)
 f / g ( x)
Example: f (x) =2x, g (x ) =x2+1
Sum:
Difference:
Product:
Quotient:
III.
Domain?
(-,)
(-,)
(-,)
(-,)
(denominator
never 0)
f (x ) + g (x ) = 2 x  x  1
2


f (x ) - g (x ) = 2 x  x 2  1 = 2 x  x 2  1


f (x ) * g (x ) = 2 x x  1  2 x 3  2 x
f (x ) / g (x ) = 2 x 2
x 1
2
The domain of Arithmetic Combinations
When two functions f and g work together, the domain of the combination function will
be values of x that are in both the domain of f AND the domain of g.
domain of a combination function = (domain of f)  (domain of g)
Also, the combination function may be undefined for new points: in the case of f/g, points
which make the denominator 0.
Example
f ( x) 
x 1
g ( x)  3  x
f ( x)  g ( x) 
x 1  3  x
domain:
x≥-1
[-1,)
x3
(-,3]
[-1,3]
how:
where x  1  0
where 3  x  0
use number line to find
intersection of the two
domains
f ( x)  g ( x) 
x 1  3  x
f ( x)  g ( x)  x  1 3  x 
[-1,3]
x  13  x
[-1,3]
f ( x)  g ( x)  3  2 x  x 2
f ( x) / g ( x)  x  1 / 3  x 
x 1
3 x

[-1,3)
also consider that the
denominator cannot be 0,
which means that x3
Example
domain:
x0
(-,0) (0,)
x2
(-,2) (2,)
x0 AND x2
(-,0) (0,2)
(2,)
1
f ( x) 
x
g ( x) 
x
x2
f ( x)  g ( x) 
1
x

x x2
how:
denominator
can’t be 0
denominator
can’t be 0
use number line to
find intersection of
the two domains
(…just showing how you could get this
answer another way)
1
x
1 1
x 1 1 x2
x x

  
 

x x2 x 1 x2 1 x x2 x2 x
1 x2
x x
( x  2)
x( x)



x x  2 x  2 x x( x  2)  x  2 x

( x  2)  x( x) x  2  x 2

x( x  2)
x( x  2)
f ( x)  g ( x) 
f ( x)  g ( x) 
1
x

x x2
1
x
x


x x  2 x x  2 
Simplify?
x
1
1


only if x0
x x2 x2
… but x0 since domain excludes x=0
1
so f ( x)  g ( x) 
x  2
1
x
1 x2 x2
f ( x) / g ( x)  
 
 2
x x2 x
x
x
when is
x ( x  2) =0?
when x=0 and
when x=2
domain does not
include x=0 and
x=2
domain:
x0 AND x2
(-,0) (0,2) (2,)
how:
use number line
to find
intersection of
the two domains
x0 AND x2
(-,0) (0,2) (2,)
x0 AND x2
(-,0) (0,2) (2,)
also consider
that the
denominator
cannot be 0,
which means
that x0, adds no
new problems
Example
domain:
(-,)
 1 if x  0
f ( x)  
 1 if x  0
1 if x  0
g ( x)  
0 if x  0
 1  1  0
f ( x)  g ( x)  
 1 0  1
 1  1  2
f ( x)  g ( x)  
 1 0  1
how:
union of (-,0) and [0, )
(-,)
if x  0
if x  0
if x  0
if x  0
(-,)
(-,)
 1  1  1 if x  0
f ( x)  g ( x)  
 1  0  0 if x  0
(-,)
 1 / 1  1 if x  0
f ( x) / g ( x)  
if x  0
 1/ 0  
(-,0)
Example
x 2  4 x  2x  2 x  2
x  2  x  2 IF x2


x2
x2
x2
Therefore, the difference between f(x) and g(x) is that f(2) is not defined but g(2) is.
f ( x) 
IV.
Skill Drill: Arithmetic Combinations of Rational Functions
Use the worksheet to make sure you know the rules for adding, subtracting, multiplying
and dividing functions.