Combinations of functions
Example
f x   x
x 
g x   x 2  1
x 
then we could define functions like
f x   g x 
f x   g x 
f x .g x 
Composite Functions
We define the composite function fg(x) or fog(x) to be “g first
followed by f”
g(x)
x
fg(x)
Example
If f(x) = 2x – 3 and g(x) = x2 + 1
Then find
(i)
fg(x)
(ii)
gf(x)
(iii)
f o f(x)
(i)
 
 2x  1  3
fg(x)  f x 2  1
2
 2x2  2  3
 2x2 1
(iii)
fof(x)  f 2 x  3
 22 x  3  3
 4x  6  3
 4x  9
(ii)
gf(x)  g 2 x  3
 2 x  32  1
 4x2  6x  9 1
 4 x 2  6 x  10
Example
The function f(x) = 3 – 2x for x ≥ 1 and g(x) = 4x – 1 for x ≥ 2
(i) Find the range of values for f(x) and g(x)
(ii) Obtain g o f(x) and obtain the range and domain of g o f(x)
(iii) Obtain fg(x) and obtain the domain and range of fg(x)
(i)Range
f(x)  1
g(x)  7
(ii) g o f(x)  g 3  2 x 
 43  2 x   1
 11  8 x
f( 1 )  1
As f(x) produces a range not acceptable for g(x)
g o f(x) is not defined.
(iii)
fg(x)  f 4 x  1
 3  24 x  1
 5  8x
Domain [2,)
Range
(, –11]
x≥ 2
fg(x) ≤ -11
Example
The functions f and g are defined with their respective domains by
f x   2  x 4
for all real values of x
g x  
for real values of x, x  4
1
x4
(a) State the range of f .
(b) Explain why the function f does not have an inverse.
(c) (i) Write down an expression for fg(x) .
(ii) Solve the equation fg(x) = 14 .
(a)
f x   2  x
y
4
2
Range (, 2]
0
x
f x   2
(b) Has no inverse as the function is a “many to one”
(c) (i) fg  x   f  1 
(ii)
 x4
 1 
 2

 x4
1
 2
x  44
4
fg  x   14
1
2
 14
4
x  4
1
16 
x  44
1
x4 
2
1
1
x4 , x3
2
2
Example
The function f(x) = x2 for x≥2 , g(x) = – x for x≥0 , h(x) = 3x – 2
1
for x≥2 and r(x) =
for x> –1
x 1
Sketch the following, giving the domains and range
(i) fg(x)
(ii) gof(x)
(iii) rf(x)
(iv) roh(x)