1.4 Notes: Building Functions from Functions
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Example 1: Find formulas for the functions f+g, f-g, and fg. Give the domain of each.
f(x) = (x-1)2
g(x) = 3-x.
Example 2: Find formulas for f/g, g/f. Give the domain of each.
a. f(x) =
x  2 g(x) =
x4
b. f(x) = x3 g(x) = 3 1  x 3
Example 3: f(x) = x 2  1 g(x) = x , Find f(g(x)) and its domain.
Example 4: Find ( g f )( x) and ( g  f ) (-2).
Find the domain of ( g f )( x) .
x
f (x) =
g(x) = 9 – x2
x 1
Example 5: Find f(x) and g(x) so that the function can be described as y = f(g(x)). (There may be
more than one decomposition.)
a. y = (x3+1)2
b. y = esinx
c. y = (tanx)2+1
Example 6: Modeling with Function Composition
In the medical procedure known as angioplasty, doctors insert a catheter into a heart
vein (through a large peripheral 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. (See Figure 1.58.)
(a) Find the volume after t seconds
(b) When the volume is V, what is the radius r?
(c) Write an equation that gives the radius r as a function of the time. What is the
radius after 5 seconds?
Implicitly defined Functions
Look at the equation of a circle, for example, x2+y2=4. While it is not a function itself, we can
split it into two equations that do define functions as follows.
x2+y2=4
y2= - x2+4
y= 
 x2  4
and y = -
 x2  4
The graphs of these two functions are, respectively, the upper and lower semicircles of a circle.
Since all the ordered pairs in either of these functions satisfy the equation x2+y2=4, we say that
the relation given by the equation defines the two functions implicitly.
Example 7: Find two functions defined implicitly by the given relation.
a. x + y2 = 25
c. x2  2 xy  y 2  1
b. x - y =1