Compound Functions

Mathemeatics lesson, subject of functions. Finding the composite function. Calculations in compound functions. Compound of two functions. Subject expression and solved questions.



How is Found the Composition of Two Functions

The composition of two functions is found by written the second function in place of x in the first function.

That is, the second function is placed inside of first function.


Example:

The first function is

f(x) = 5x + 1

The second function is

g(x) = 4x + 3


The composition of two function.

fog(x) = 5(4x + 3) + 1

fog(x) = 5(4x + 3) + 1

fog(x) = 20x + 15 + 1

fog(x) = 20x + 16

In the example, we wrote g (x) instead of x in the function f (x).

The composition of two function is shown as the following.

fog(x) = f(g(x))

If the "g" function is inside of the "f" function, the composition of f and g is shown as

fog = f(g(x))


If the "f" function is inside of the "g" function, the composition of f and g is shown as

gof(x) = g(f(x))

Example:

f(x) = 7x – 3 

g(x) = 6x + 5

Find the "fog(x)" and "gof(x)" functions.


Solution:

The function "fog(x)" is found by writing the "g(x)" function in place of x in the "f(x)" function.

fog(x) = f(g(x))

f(g(x)) = 7(6x + 5) – 3 

f(g(x)) = 42x + 32


The function "gof(x)" is found by writing the "f(x)" function in place of x in the "g(x)" function.

gof(x) = g(f(x))

g(f(x)) = 6(7x – 3) + 5

g(f(x)) = 42x – 13


Compound Functions Defined on Finite Sets

A, B and C is three sets. F(x) and g(x) is two functions.

f: A → B

g: B → C

In this case,

gof: A → C 


In the composite function, the elements of set A matches with the elements of set C, using g function.

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Example:

f(x) = 4x + 11

g(x) = 6x – 2


Let we examine x =3 value in the f(x), g(x), gof(x) and fog(x) functions.


f(3) = 4*3 + 11

f(3) = 23

Let we write the 23 in place of x in g(x) function.

g(23) = 6*23 – 2

g(23) = 136


Let we find gof(x) function

gof (x) = 6(4x + 11) - 2

gof (x) = 24x + 66 – 2 

gof (x) = 24x + 64


Now, Let we write x = 3

gof (3) = 24*3 + 64

gof (3) = 136


As seen at the example, the gof(x) paires the 3 value to the 136 value.

Let we examine fog(x) too.

g(3) = 6*3 – 2

g(3) = 16


f(16) = 4*16 + 11

f(16) = 75


fog(x) = 4(6x – 2) + 11

fog(x) = 24x + 3

fog(3) = 75

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The function fog(x) is shown in the form fog(x) or f((g(x)).


Example:

f(x) = x3 + 1

g(x) = x + 2

Determine the fog(x).



Solution:

The function fog(x) is found by writing (x + 2) in place of x in the (x3 + 1) function.


fog(x) = (x + 2)3 + 1

fog(x) = x3 + 6x2 + 12x + 8 + 1

fog(x) = x3 + 6x2 + 12x + 9


Example:

f(x) = 2x2 + 5x + 1

g(x) = x – 3

Find the fog(x) function.


Solution:

fog(x) = f[g(x)]

f[g(x)] = 2*(x – 3 )2 + 5(x – 3) + 1

f[g(x)] = 2(x2 – 6x + 9) + 5x – 15 + 1

f[g(x)] = 2x2 – 12x + 18 + 5x – 15 + 1

f[g(x)] = 2x2 – 7x + 4


Example:

f(x) = 3x2 + 10

g(x) = 4x – 6

Find gof(2) value.


Solution:

gof(x) = 4*(3x2 + 10) – 6

gof(x) = 12x2 + 40 – 6

gof(x) = 12x2 + 36


For x = 2,

gof(2) = 12*22 + 36

gof(2) = 84


Example:

f(x) = 5x – 1 

g(x) = 6x + 7


gof(6) değerini bulunuz.


Solution:

Method 1

gof(x) = 6(5x – 1) + 7

gof(x) = 30x + 1

gof(6) = 30*6 + 1

gof(6) = 181



Method 2

We writes (5x – 1) in place of x in the g(x) function.

gof(x) = g[f(x)]

g(5x – 1) = 6x + 7

g(5*6 – 1) = 6*29 + 7 

g(29) = 181



Example:

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What's the fog(2) + gof(4) sum?


Solution:

fog(2) = f[g(2)]

Let we find g(2)

For x = 2, g(x) = x2 – 3 

g(2) = 22 – 3 

g(2) = 1


For x = 1, f(x) = x – 2 

f(1) = 1 – 2 

f(1) = - 1

fog(x) = - 1 


Let we find gof(4) function.

For x = 4, f(x) = 3x + 1 

f(4) = 3*4 + 1

f(4) = 13


For x = 13, g(x) = 4x – 5 

g(13) = 4*13 – 5 

g(13) = 47


gof(4) = 47

fog(2) + gof(4) = - 1 + 47

= 46


Function Concept In Mathematics

Operations On Functions

Inverse Functions


RISE KNOWLEDGE

15/08/2018

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