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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.

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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

9

25

49

1

2

3

4

1

3

5

7

Range of f Domain of g

Domain of f

Range of g

fg and gf
fg and gf

In general, the composite function fg and gf are different functions

f(x) = 2x – 1 and g(x) = x2

gf(x)

2nd function applied

1st function applied

gf(x) = (2x – 1)2

e.g. gf(3) = 25

e.g. fg(3) = 17

fg(x) = 2x2 - 1

examples
Examples

Find f(3) and f(-1)

f(3) = (43 – 1)2 = 121

f(-1) = (4-1- 1)2=(-5)2= 25

Find (i) gf(2) (ii) gg(2) (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

examples1
Examples

Break the following functions down into two or more components.

(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) Domain: x  2 range f(x)  2

(ii) Domain: x  0 range f(x)  0

examples2
Examples

Express the following functions in terms of f, g and h as appropriate.

  • 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
Inverse functions

The inverse function of f maps from the range of f back to the domain.

f has the effect of ‘double and subtract one’ the inverse function (f -1) would be ‘add one and halve’.

f(x)

range of f

domain of f -1

domain of f

range of f -1

A

B

f -1(x)

The inverse function f -1 only exists if f is one – one for the given domain.

graph of inverse functions

y

x

Graph of inverse functions

f(x) = 2x - 1

f(2) = 3  (2, 3)

y = x

f -1(3) = 2  (3, 2)

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
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

find the inverse f - 1 (x).

examples3
Examples

find the inverse f - 1 (x).

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

examples4
Examples

find the inverse f - 1 (x).

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