Onward to Section 1.4a…
Combining functions
algebraically, composite
functions, and decomposing
functions!
Definition: Sum, Difference, Product, and Quotient of
Functions
Let f and g be two functions with intersecting domains. Then for
all values of x in the intersection, the algebraic combinations of
f and g are defined by the following rules:
Sum:
Difference:
Product:
Quotient:
 f  g  x   f  x   g  x 
 f  g  x   f  x   g  x 
 fg  x   f  x  g  x 
f  x
 f 
, provided g  x   0
  x 
g  x
g
In each case, the domain of the new function consists of all
numbers that belong to both the domain of f and the domain of g.
As noted, the zeros of the denominator are excluded from the
domain of the quotient.
Guided Practice
For the given functions, find f + g, f – g, fg, f/g, and gg. Give
the domain of each.
f  x  x
2
g  x  x 1
First, find the domain of the original functions, and determine
where these two domains intersect (overlap).
Domain of f:
 ,  
Domain of g:
Domain intersection:
1, 
1, 
This intersection becomes the domain of all of the algebraic
combination functions!!!
Guided Practice
For the given functions, find f + g, f – g, fg, f/g, and gg. Give
the domain of each.
f  x  x
2
g  x  x 1
 f  g  x   f  x   g  x   x  x  1
2
with D:
1, 
 f  g  x   f  x   g  x   x  x  1
2
with D:
1, 
 fg  x   f  x  g  x   x x  1
with D:  1,  
2
Guided Practice
For the given functions, find f + g, f – g, fg, f/g, and gg. Give
the domain of each.
f  x  x
2
f  x
 f 

   x 
g  x
g
g  x  x 1
2
x
x 1
 gg  x   g  x  g  x   
x 1
with D:

 1, 
2
with D:
1, 
Can we simplify this last one???
Guided Practice
For the given functions, find formulas for the functions f + g,
f – g, and fg. Give the domain of all functions.
f  x  x  5
D :  5,  
g  x  x  3
D :  ,  
 f  g  x   x  5  x  3
 f  g  x   x  5  x  3
 fg  x   x  3 x  5
Domain of all 3 combination functions: 5,  
Guided Practice
For the given functions, find formulas for f/g and g/f. Give the
domain of all functions.
f  x  x  2
D :  2,  
g  x  x  4
D :  4,  
 f 
  x 
g
x2

x4
x2
D :  2,  
x4
g
  x 
 f 
x4

x2
x4
D :  2, 
x2
Now on to
composite
functions?!
Definition:
Composition of Functions
Let f and g be two functions such that the domain of f intersects
the range of g. The composition of f and g, denoted f g, is
defined by the rule
f
g  x   f  g  x  
The domain of f g consists of all x-values in the domain of g
that map to g(x)-values in the domain of f.
NOTE: In most cases, f g and g f are different functions!!!
A Few Practice Problems…
For the given functions, find (f g)(x) and (g f)(x) and verify
(both algebraically and graphically) that the two composite
functions are not the same.
f  x   ex
f
g
g  x  x
 x  e
f  x   g  f  x    g  e   e
g  x   f  g  x    f
x
Now, how do we verify???
x
x
A Few Practice Problems…
For the given functions, find (f g)(x) and (g f)(x) and give the
domain of each composition function.
f  x   x2 1 g  x   x
f
g  x  
 g f  x 
 x
 x 1
2
2
1
D:
D:
0,
 , 1 1,  
Let’s check these with the calculator…
A Few Practice Problems…
For the given functions, find (f g)(3) and (g f)(–2).
f  x   x 1 g  x   2 x  3
2
f
g  3  f  g  3 
8
g
f  2  g  f  2 
3
Decomposing Functions…working
backwards or undoing a composition…
For each function h, find functions f and g such that h(x) = f(g(x)).
h  x    x  1  3  x  1  4
2
f  x   x  3x  4
2
g  x  x 1
Decomposing Functions
For each function h, find functions f and g such that h(x) = f(g(x)).
h  x  x 1
3
f  x  x
g  x  x 1
3
Any other ways to solve this one?!?!
Let’s do some modeling…
In math-land, not fashion-land…
In the medical procedure known as angioplasty, doctors insert a
catheter into a heart vein and inflate a small, spherical balloon
on the tip of the catheter. Suppose the balloon is inflated at a
constant rate of 44 cubic millimeters per second.
1. Find the volume after t seconds.
V = 44t
Let’s do some modeling…
In math-land, not fashion-land…
In the medical procedure known as angioplasty, doctors insert a
catheter into a heart vein and inflate a small, spherical balloon
on the tip of the catheter. Suppose the balloon is inflated at a
constant rate of 44 cubic millimeters per second.
2. When the volume is V, what is the radius r ?
4 3
πr  V
3
3V
r 
4π
3
3
V
r3
4π
Let’s do some modeling…
In math-land, not fashion-land…
In the medical procedure known as angioplasty, doctors insert a
catheter into a heart vein and inflate a small, spherical balloon
on the tip of the catheter. Suppose the balloon is inflated at a
constant rate of 44 cubic millimeters per second.
3. Write an equation that gives the radius r as a function of the
time. What is the radius after 5 seconds?
V  44t
and
3
V
r3
4π
3
44
t


33
t
3
r
3
4π
π
At 5 seconds,
r = 3.745 mm
Whiteboard Practice
For the given functions, find formulas for the functions f + g,
f – g, and fg. Give the domain of all functions.
f  x    x  1
2
g  x  3  x
2
x
 3x  4
f

g
x


 
2
x
 x2
f

g
x


 
fg ( x )   x 3  5 x 2  7 x  3
Domain of all five functions:
 ,  
Whiteboard Practice
Find f(g(x)) and g(f(x)). State the domain of
each.
1
2
g ( x) 
f ( x)  x  1
x 1
1
f ( g ( x )) 
1
2
( x  1)
D : ( ,1)  (1,  )
1
x2  2
D : ( ,  2)  (  2, 2)  ( 2, )
g ( f ( x )) 
Whiteboard Practice
Find f(x) and g(x) so that the function can be
described as y=f(g(x)).
y  ( x  1)
3
2
f ( x )  ( x  1)2
g ( x)  x
3
Homework: p. 127-128 1-23 odd
one possible
solution…