1.5 COMBINATIONS OF FUNCTIONS
You should learn to:
1. Add, subtract, multiply and divide functions.
2. Find compositions of one function with another function.
3. Use combinations of functions to model and solve real-life problems.
Terms to know: sum, difference, product, and quotient (arithmetic combinations of functions),
composition. of functions
1. Sum:
(f + g)(x) = f(x)+ g(x)
2. Difference:
~ - g)(x) = J(x) -
3. Product:
(Jg)(x) = f(x)· g(xL
4. Quotient:
( f)(x)J(x), g(x)" 0
g
g(x)
Example': Given
J(x) = x2 -4 -~.d g(x) = x-3, fmd:
b. (g-f)(x)
a.
(J+g)(x)
g(x)
l; ~J-
x-3- c~'t- ~
f~~1 r ,"t.J 1'trn. ~
c.
(fg)(x)
)
')(. - 'S - )(.L. + L(
(
d.
- 'J.'t t- X
---~-)
')(-3
+/2
+ ']
e
l-})(x)
-~;~~~-~3~---------X 1-_ '-(
"-- f !: (__
)
l
Which of the functions in parts a- e have domain restrictions? List the domain restrictions.
55
Example 2: Using the functions above, fmd each of the following.
a.
-=+(1)+
(/+g)(3)
~J
b. (/- g)(2)
F(-z) - <J(z)
0
d.
c. (fg)(-2)
~ (-'L) • !3 (- z)
D- - /
e. (~)(-~
(~}4)
1@
[D
L§J
When combining functions using addition, subtraction, or multiplication, whatever domain
restrictions occur
in the components (parts) will occur in the combined functions.
.
...1 7
0
'f..
~1
Example3: Given f(x)=£ and g(x)=- , fmd (f-g)(x) and (g[)(x) , andlistthe
rD
-
domain restrictions for each.
a.
(/ -
X
\.R -.J...
g)(x)=
X
b.
-
l ..
(g[)(x) =
Ux
D:
When dividing functions, all domain restrictions occurring in the original functions will occur in
the combined function. However, new domain restrictions may occur (mainly because the
0
X !::: ;;;<
_X
denominator ofthe new function cannot equal zero).
2
Example 4: Given
)ex)
I
a. ( 1L
f (x) = ~)2- x
~
vz-x-
b.
e
(~)(x)
L.}
-y.~'Z.
y..i- 0
~
.:..----( : rJO 1
56
o)u ( o t
}1)v:Z-i
_!__.
(1L
f
f.
(~}!)
t!p'
[j]
'/..
0
x, find:
c.
-x.:?-2.
(~ )(o) ~
Jz._ C>
ill
)(.LL
( - c*>,
and g( x) =
-z
[§]
g.
(~}o)
(~ }3)
d.
(\..C
3
~
so/
+
o( (.) 1\'l(l i'l-1
t\..0
i'Y\
h. (~)(3)
~ ~61]
L-:;:Jl
s ~~ ~ o(~
A4 omgosition of functiofiJoccurs when one function is "put into" another function. The
compositiOn
of function f with function g is symbolized by (/ o g)(x) or f(g(x)) and means that g(x)
:~utinto" ·f(x).
ccl'Vtpo.-.,e '..
c_ht¢ -res\d'c..h~s : <:::,lr\1\p} : fd
g(f(x))=
~f)c,)=
Example 5: Given
each function.
__a,.
f (x)
l(f_o_g_)(_x)----~~~-/?0
(?
:X. -l
3
Lft.z.
t"
'2._
J2X.. t-~
md the following. List the domain for
b. (g o f)(x)
tlX -I
Vx- t
x'D
[b) gp)
._..·
Example 6: Given f(x)
a. ~?)(x)
(go /)(x)
. ·z(x-~+ I
'2.)(. - ~T-/
[ Lx
-·7]
•
c.
f(g(3))
~
13]
Example 7: Given
a. f(g(x))
(!X)1_
- L><-
nd:
2j.. -\ 1- <-t
..
(u.-?J
~
d. g(/(-1))
EfJ
f (x)
b. g(f(x))
JXL
x"L:? 0
~~~
)C.: ~0
+
•c +-
t ~jcP)
\ X\
57
~~omposite functions on your calculator to "s~e" their domains. Hint: ~et
~ ~~d find
Y3 = Y1 (Y2 ) and Y4 =Y2 (y1 ) usmg the V~menu. (Hint:
Make sure to use parentheses, or you will be graphing products- instead of compositions.) What
did you notice?
.
Make sure to look at restrictions in domains both before and after forming compositions.
To "decompose" a function, look for "inner'' and "outer" functions. There are oftentimes more
'
than one way to "decompose" a function.
Example 9: Write each function as the composition of two functions. That is, h ( x) = f ( g ( x)),
where _ _ __
a.
fh{x)=~l
g(x)=
_i__
x'L- "2-
f(x)=
~
[b. h(x)=.J2x-3 J
g(x)= 2,\-3
-
-
l
~
x -L..
.
Example 10: The cost of producing x units in a manufacturing process is given by th
___, C x =48 + 1150 . The nutp.ber ~~ units produced in t hours__is given by x t
40t .
~
a. Find and interpret Cox
({_ o
=
I q 20
b
-r ll sa
+- (/ .s- 0
b. Find x(20) and interpret
1-. (
t-)
<aDO ur\l' rs
~ l{, 0 t
)(. c-zfi) ::- goo
c. Find
(Cox )('20)
- tV"
LUe\1\
i
hO\J'fS.
prduce~
L..o ~rs.
and interpret.
";>.
-
lctd..o
t + uS6
tq2o(~ + rts-o
- 1l
58
l: o.s 4- o~ un~d s
p rooluc-ecl i 11
x'y~) ~ ll8' ('-lot)
3Cf,ss-D
~ -$! 3q~s-~o 1-o ~td\tt'
~0 un (d~
IY\
a.() ATS.