Worksheet 2: Composition of functions
Name:
Date:
Let there be two functions defined as:
f:AB by f(x) for all xA
g: BC by g(x) for all xB
Then, the new function, gof
read as "g circle f" or "g
composed with f", is defined
as:
g o f ( x) g ( f ( x)) , for all xA
Domain of
f(x)
Domain of
Range of f(x)
Domain of
g(f(x))
Range of g(x)
Range of h(x) =
g(f(x))
Example 1:
Let two functions be defined as:
f={(1,2) ,(2,3) ,(3,4) ,(4,5) } and g={(2,4) ,(3,2) ,(4,3) ,(5,1) }
Check whether gof and fog exist for the given functions.
Solution:
f
g
Domain
{1,2,3,4}
{2,3,4,5}
Range
{2,3,4,5}
{4,2,3,1}={1,2,3,4}
Hence,
Range of fDomain of g gofexists.
Range of gDomain of f fog exists.
It means that both compositions gof and fog exist for the given sets.
Example 2:
2
Given f ( x ) 2 x 1 and g ( x) x 3 , find:
a. ( f o g)(x).
b. ( g o f)(x).
c. ( f o f)(x).
d. ( g o g)(x).
Solution:
f og x f g x f x 2 3 2 x 2 3 1 2 x 2 7
a.
.
2
g o f x g f x g 2 x 1 2 x 1 3
b.
4 x 2 4 x 1 3 4 x 2 4 x 2
c.
d.
f o f x f f x f 2 x 1 2 2 x 1 1 4 x 3
g og x g g x g x 2 3 x 2 3
x4 6 x2 9 3 x4 6 x2 6
Note that:
f og x 2 x 2 7 g o f x 4 x 2 4 x 2
That is, ( f o g)(x) is not the same as (g o f )(x). The open dot "o" is not the same as a
multiplication dot "", nor does it mean the same thing.
f(x) g(x) = g(x) f(x)
[always true for multiplication]
...you cannot say that:
( f o g)(x) = (g o f )(x)
[generally false for composition]
Domain and range of the composition of functions
Consider the function:
f ( x)
1
,
1 x when x 1 Domain of f is x : x R \ 1 , i.e. all real numbers except 1.
Let us now see the expression of composition of function with itself,
1
1
1
1 x
1
1 x
1
1
1 x x 1
1 x
1
x
x
x
1 x 1 x 1 x
f o f x f f x f
valid for real values of x0.
Since f is undefined for x = 1, and f o f is undefined for x = 0, thus the domain of the composition
f o f x is : x : x R \ x 0, x 1 ; i.e. all real numbers except 0 and 1.
Sometimes you have to be careful with the domain and range of the composite function.
General rule to determine the domain:
f ( x)
Polynomial
Domain
f ( x)
x0
1
x
f ( x) x
f ( x) log( x)
f ( x) a x
x0
x0
, for a 0 ,
0
,for a 0
Example:
Given f ( x ) x and g ( x) x 3 , find the domains of ( f o g)(x) and (g o f )(x).
Solution:
f ( x) x x 0
So:
f og x f g x x 3 x 3 0 x 3
Hence, the domain of ( f o g)(x) is "all x > 3".
Now do the other composition:
g o f x g g x ...
Hence, the domain of (gof)(x) is
Going backward: given composed function, find original functions
Usually composition is used to combine two functions. But sometimes you are asked to go backwards.
That is, they will give you a function, and they'll ask you to come up with the two original functions
that they composed.
Example 1:
h x x 5 3 x 5 7
Given
, determine two functions f (x) and g(x) which, when composed,
generate h(x).
2
Solution:
This is asking you to notice patterns and to figure out what is "inside" something else.
2
In this case, this looks similar to the quadratic x 3 x 7 , except that, instead of squaring x, they're
squaring x + 5.
f x x 2 3x 7
So let's make g(x) = x + 5, and then plug this function into
:
2
f og x f g x f x 5 x 5 3 x 5 7
Then h(x) may be stated as the composition of
f x x 2 3x 7
and g(x) = x + 5.
Example 2:
Given
h x 3x 4
, determine two functions f (x) and g(x) which, when composed, generate h(x).
Solution:
Since the square root is "on" (or "around") the "3x + 4", then the 3x + 4 is put inside the square root,
that is:
h ( x ) f ( g ( x ))
g x
x 3x 4
Thus, g(x) = 3x + 4,
f x
f x x
3x
, and h(x) = ( f o g)(x).
Exercise
For the given functions:
a. f(x) = x + 1 , g(x) = 3x
2
b. f ( x) x 1, g ( x ) 2 x
2
c. f ( x) x 1, g ( x) x 5
d. f(x) = 2x + 1 , g(x) = x2
Find:
1. Domain and range of each f(x) and g(x)
f x
g x
a. Domain =
Domain =
Range =
Range =
b. Domain =
Domain =
Range =
Range =
b. Domain =
Domain =
Range =
Range =
c. Domain =
Domain =
Range =
Range =
2. Determine
f og x
a.
f og x
and its domain
Domain =
b.
f og x
Domain =
c.
f og x
Domain =
d.
f og x
Domain =
a.
3. Determine
g o f x
g o f x and its domain
Domain =
b.
g o f x
Domain =
c.
g o f x
Domain =
d.
g o f x
Domain =
4. Determine
f o f x
a.
f o f x and its domain
Domain =
b.
f o f x
Domain =
c.
f o f x
Domain =
d.
f o f x
Domain =
5. Determine
g og x
a.
g og x and its domain
Domain =
b.
g og x
Domain =
c.
g og x
Domain =
d.
g og x
Domain =
f ( x)
1
1 x for all real values except x =1 .
6. A function is defined for real values by :
f f f x
Determine
and draw the graph of resulting composition!
7. Given f(2) = 3, g(3) = 2, f(3) = 4 and g(2) = 5, evaluate (f o g)(3)!
8. Functions f and g are as sets of ordered pairs
f = {(-2,1),(0,3),(4,5)} and g = {(1,1),(3,3),(7,9)}
Find the composite function defined by g o f and describe its domain and range.
9. Write function F given below as the composition of two functions f and g, where
F ( x)
g ( x)
1
x and
1 x
10. Evaluate f(g(h(1))), if possible, given that
11. For the composite function
f og ( x ) x , f ( x ) x 2 2
a.
b.
c.
d.
f og ( x )
h( x) x , g ( x) x 1,
f og ( x ) 2 x 6 x 2 1 f ( x ) 2 x 3 x 1
,
f og ( x ) x 1 4
f og ( x ) x, f x x 2 5
1
x2.
g x
and f ( x) , find
!
and
f ( x)
f x x2 4
f og ( x) and g ( x) , find f x !
12. For the composite function
f og ( x) sin x 2 1 g ( x) x 2 1
e.
,
f.
f og ( x ) x ,
g.
h.
g ( x) 1 x 2
f o g ( x ) 4 x, g ( x ) x
f og ( x )
1
x
1
1
1 1 g ( x) 1
x
x
,
10
