Composite functions

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# Composite functions - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Composite functions' - paula-harris

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Presentation Transcript
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

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

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

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

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

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.

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

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).

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

Examples

find the inverse f - 1 (x).