Functions - Algebra

* Operations of Functions

(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - g(x)

(f . g)(x) = f(x) . g(x)

                                    Note: g(x) is not equal to zero

Where: 

f(x) and g(x) are functions and read as "f of x and g of x."

Example:

Given:

f(x) = 3x + 2
g(x) = 5x - 4

Required: (f + g)(x), (f - g)(x), (f . g)(x), (f/g)(x)

Solution:
* (f + g)(x) = f(x) + g(x)

= 3x + 2 + (5x - 4)

= 3x + 2 + 5x - 4

= 8x - 2

* (f - g)(x) = f(x) - g(x)

= 3x + 2 - (5x - 4)

= 3x + 2 - 5x + 4

= -2x + 6

* (f . g)(x) = f(x) . g(x)

= (3x + 2)(5x - 4)  by using the FOIL method

= (3x)(5x) + (3x)(-4) + (2)(5x) + (2)(-4)

= 15x² - 12x + 10x - 8

= 15x² - 2x - 8

 

* Compositions of Functions

f[g(x)] = fog composition of f of g on x
g[f(x)] = gof composition of g of f on x


Examples:

Given:

f(x) = 3x - 2
g(x) = 2x + 3

Required: fog and gof

Solution: 

* fog
f[g(x)] = 3(2x + 3) - 2

= 6x + 9 - 2

= 6x + 7

* gof
g[f(x)] = 2(3x - 2) + 3

= 6x - 4 + 3

= 6x - 1

Given:

f(x) = x² - 3
g(x) = x + 2

Required: fog and gof

Solution:

* f[g(x)] = (x + 2)² – 3 by using Square of Binomials

= x² + 4x + 4 - 3

= x² + 4x + 1

* g[f(x)] = (x² - 3) + 2

= x² - 1

- it is just like substituting the value of one function inside the other.

* Inverse of a Function

To find the inverse of a given function we need to follow these 4 steps:

1. Let f(x) be equal to y
2. Interchange x and y and vise-versa.
3. Solve for y. 
4. Change y to f^-1(x).

Examples: 

Given:

f(x) = 4x - 5 

Required: f^-1(x)

Solution: 

Step 1:

f(x) = 4x - 5

y = 4x - 5

Step 2:

y = 4x - 5

x = 4y - 5

Step 3:

x = 4y - 5
4y = x + 5

Step 4:

Checking: 

f[f ^-1(x)] = x

x + 5 - 5 = x
x = x


Note:
- If we get the composition of a function and its inverse, the answer will be equal to x. This can be used for checking if the inverse that we have solved is correct.