# 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.

**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

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

**Example:**

f(x) = x^{3} + 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 (x^{3} + 1) function.

fog(x) = (x + 2)^{3} + 1

fog(x) = x^{3} + 6x^{2} + 12x + 8 + 1

fog(x) = x^{3} + 6x^{2} + 12x + 9

**Example:**

f(x) = 2x^{2} + 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(x^{2} – 6x + 9) + 5x – 15 + 1

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

f[g(x)] = 2x^{2} – 7x + 4

**Example:**

f(x) = 3x^{2} + 10

g(x) = 4x – 6

Find gof(2) value.

**Solution:**

gof(x) = 4*(3x^{2} + 10) – 6

gof(x) = 12x^{2} + 40 – 6

gof(x) = 12x^{2} + 36

For x = 2,

gof(2) = 12*2^{2} + 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:**

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

**Solution:**

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

Let we find g(2)

For x = 2, g(x) = x^{2} – 3

g(2) = 2^{2} – 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**

RISE KNOWLEDGE

15/08/2018

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