Transcript Composite functions

Composite functions
When two or more functions are combined, so that the output from
the first function becomes the input to the second function, the
result is called a composite function or a function of a function.
Consider f(x) = 2x -1 with the domain {1, 2, 3, 4} and g = x2
with domain the range of f.
gf(x)
f(x) = 2x - 1
g(x) = x2
1
1
1
2
3
9
3
5
25
4
Domain of f
7
Range of f
Domain of g
49
Range of g
fg and gf
In general, the composite function fg and gf are different functions
f(x) = 2x – 1 and g(x) = x2
2nd function applied
gf(x)
1st function applied
gf(x) = (2x – 1)2
e.g. gf(3) = 25
fg(x) = 2x2 - 1
e.g. fg(3) = 17
Examples
Given f : x
( 4x 1)2 ,where x 
Find f(3) and f(-1)
f(3) = (43 – 1)2 = 121
Given f : x
x2 and g :x
Find (i) gf(2)
(ii) gg(2)
f(-1) = (4-1- 1)2=(-5)2= 25
2x 1,where x 
(iii) fg(2)
gff(2)
(i) gf(x) = 2x2 – 1  gf(2) = 222 – 1 = 7
(ii) gg(x) = 2(2x – 1)– 1  gg(2) = 2(22-1) – 1 = 5
(iii) fg(x) = (2x – 1)2  fg(2) = (22 – 1)2= 9
(iv) gff(x) = 2x4 - 1 gff(2) = 224 – 1 = 31
Examples
Break the following functions down into two or more components.
(i ) f : x
2x2  3 (ii )g : x
( x  3 )4
(i) f(x) = 2x + 3 and g(x) = x2  fg(x) = 2x2 + 3
(ii) f(x) = x , g(x) = x - 3 and h = x4  hgf(x) = (x – 3)4
Find the domain and corresponding range of each of the
following functions.
(i ) f : x
x  2 (ii ) f : x
( x  3 )4
(i) Domain: x  2 range f(x)  2
(ii) Domain: x  0 range f(x)  0
Examples
Given that f : x
x  4, g : x
3x and h : x
x2
Express the following functions in terms of f, g and h as appropriate.
(i) x  x2 + 4
(ii) x  x6
(iii) x  3x + 12
(iv) x  9x2 + 4
(v) x  (3x + 4)2
(vi) 3x + 12
(i) fh(x) = x2 + 4
(ii) hhh(x) = x6
(iii) gf(x) = 3x + 12
(iv) fggh(x) = 9x2 + 4
(v) hgf(x) = (3x + 4)2
(vi) fffg(x) = 3x + 12
Inverse functions
The inverse function of f maps from the range of f back to the
domain.
Consider f ( x )  2x 1
f has the effect of ‘double and subtract one’ the inverse function (f
-1) would be ‘add one and halve’.
f 1( x ) 
x 1
2
domain of f
range of f
-1
f(x)
range of f
A
B
domain of f -1
f -1(x)
The inverse function f -1 only exists if f is one – one for the given
domain.
Graph of inverse functions
f(x) = 2x - 1
f(2) = 3  (2, 3)
f -1(3)
= 2  (3, 2)
y
y=x
f 1( x ) 
x 1
2
x
In general, if (a, b) lies on y = f(x) then (b, a) on y = f – 1(x).
For a function and its inverse, the roles of x and y are
interchanged, so the two graphs are reflections of each other in
the line y = x provided the scales on the axes are the same.
Finding the inverse function f -1
Put the function equal to y.
Rearrange to give x in terms of y.
Rewrite as f – 1(x) replacing y by x.
Example
Given f ( x )  2x  3, x 
Let 2x  3  y
2x  y  3
y 3
x
2
x 3
1
so f ( x ) 
, x
2
find the inverse f
- 1 (x).
Examples
Given f ( x )  x2  2x, x  1
find the inverse f
x2 -2x = y
(x – 1)2 – 1 = y
(x – 1)2 = y + 1
x – 1 = (y + 1)
x = (y + 1)+ 1
f -1(x) = (x + 1)+ 1 x  - 1
- 1 (x).
Examples
2x  1
Given f ( x ) 
,x  4
x4
find the inverse f
2x  1
y
x4
2 x  1  xy  4 y
2 x  xy  1  4 y
x( 2  y )  1  4 y
1  4 y
x
2 y
1 4 y
x
y2
1  4x
f 1( x ) 
,x2
x2
- 1 (x).