Operations on Functions: Part 1 - Addition and Subtraction

 Addition of Functions
`(f+g)(x) = f(x) + g(x)`

Note: Terms can only be added or subtracted if they are similar – when they have the same variable part (literal coefficient).  For example, you can add/subtract `3x^2` and `-2x^2` because they have the same variable part `x^2` (including the exponent of the variables). You cannot add/subtract `-7x` and `4x^2` because they are not similar terms.

Given:                   `f(x) = 3x+4`                                   `g(x)= 2x-7`

1. `(f+g)(x)`

`=> (f+g)(x) = f(x) + g(x)`

`=> = (3x + 4) + (2x - 7)`

`=> = 3x + 2x + 4 - 7`

`=> = 5x - 3`

2. `(g+f)(-3)`

`=> (g+f)(-3) = g(-3) + f(-3)`

`=> = (2x - 7) + (3x + 4)`

`=> = 2(-3) - 7 + 3(-3) + 4`

`=> = -6 - 7 - 9 + 4`

`=> = -18`

Alternatively, you can solve `g(-3)` and `f(-3)` individually and then add their results.

`=> (g+f)(-3) = g(-3) + f(-3)`

for `g(-3)`

`=> g(-3)= 2x - 7`

`=> = 2(-3) - 7`

`=> = -6 - 7`

`=> = -13`

for `f(-3)`

`=> f(-3)= 3x + 4`

`=> = 3(-3) + 4`

`=> = -9 + 4`

`=> = -5`

Thus,

 `=> (g+f)(-3) = g(-3) + f(-3)`

`=>  = -13 + -5`

`=>  = -18`

Subtraction of Functions `(f-g)(x) = f(x) - g(x)`

Note: Terms can only be added or subtracted if they are similar – when they have the same variable part (literal coefficient).  For example, you can add/subtract `3x^2` and `-2x^2` because they have the same variable part `x^2` (including the exponent of the variables). You cannot add/subtract `-7x` and `4x^2` because they are not similar terms.

Given:                   `f(x) = 3x+4`                                         `g(x)= 2x-7`

3. `(f-g)(x)`

`=> (f-g)(x) = f(x) - g(x)`

`=> = (3x + 4) - (2x - 7)` 

Note: When you have a negative sign `-` before a parenthesis `(`, what you do is flip the signs of all the terms inside the parenthesis. (i.e. positive becomes negative and vice-versa.)  

`=> = 3x + 4 - 2x + 7`  (Notice how `2x` becomes `-2x` and `-7` becomes `+7`)

`=> = 3x - 2x + 4 + 7`

`=> = x + 11`

4. `(g-f)(-2)`

`=> (g-f)(-2) = g(-2) - f(-2)`

`=> = (2x - 7) - (3x + 4)`

`=> = 2x - 7 - 3x - 4`

`=> = 2(-2) - 7 - 3(-2) - 4`

`=> = -4 - 7 + 6 - 4`

`=> = -9`

Alternatively, you can solve `g(-2)` and `f(-2)` individually and then subtract their results.

`=> (g-f)(-2) = g(-2) - f(-2)`

for `g(-2)`

`=> g(-2)= 2x - 7`

`=> = 2(-2) - 7`

`=> = -4 - 7`

`=> = -11`

for `f(-2)`

`=> f(-2)= 3x + 4`

`=> = 3(-2) + 4`

`=> = -6 + 4`

`=> = -2`

Thus,

 `=> (g-f)(-2) = g(-2) - f(-2)`

`=>  = (-11) - (-2)`

`=>  = -11 + 2`

`=>  = -9`

EXERCISES

Given: `f(x) = -3x + 7`                               `g(x) = 2x + 4`

1. `(g+f)(x)`
   
   `=> (g+f)(x) = g(x) + f(x)`
   `=> = (2x + 4) + (-3x + 7)`
   `=> = 2x - 3x + 4 + 7`
   `=> = -x + 11`
2. `(f+g)(-2)`
   
   `=> (f+g)(-2) = f(-2) + g(-2)`
   `=>  = f(-2) + g(-2)`
   
   For `f(-2)`
   `=> f(-2) = -3x + 7`
   `=> = -3(-2) + 7`
   `=> = 6 + 7`
   `=> = 13`
   
   For `g(-2)`
   `=> g(-2) = 2x + 4`
   `=> = 2(-2) + 4`
   `=> = -4 + 4`
   `=> = 0`
   
   `=>  = f(-2) + g(-2)`
   `=>  = 13 + 0`
   `=>  = 13`
3. `(f+g)(x+7)`
   
   `=> (f+g)(x+7) = f(x+7) + g(x+7)`
   
   For `f(x+7)`
   `=> f(x+7) = -3x + 7`
   `=> = -3(x + 7) + 7`
   `=> = -3x - 21 + 7`
   `=> = -3x - 14`
   
   For `g(x+7)`
   `=> g(x+7) = 2x + 4`
   `=> = 2(x+7) + 4`
   `=> = 2x + 14 + 4`
   `=> = 2x + 18`
   
   `=>  = f(x+7) + g(x+7)`
   `=>  = -3x - 14 + 2x + 18`
   `=>  = -x + 4`
4. `(g-f)(3)`
   
   `=> (g-f)(3) = g(3) - f(3)`
   
   For `g(3)`
   `=> g(3) = 2x + 4`
   `=> = 2(3) + 4`
   `=> = 6 + 4`
   `=> = 10`
   
   For `f(3)`
   `=> f(3) = -3x + 7`
   `=> = -3(3) + 7`
   `=> = -9 + 7`
   `=> = -2`
   
   `=>  = g(3) - f(3)`
   `=>  = 10 - (-2)`
   `=>  = 10 + 2`
   `=>  = 12`
5. `(f-g)(2x)`
   
   `=> (f-g)(2x) = f(2x) - g(2x)`
   
   For `f(2x)`
   `=> f(2x) = -3x + 7`
   `=> = -3(2x) + 7`
   `=> = -6x + 7`
   
   For `g(2x)`
   `=> g(2x) = 2x + 4`
   `=> = 2(2x) + 4`
   `=> = 4x + 4`
   `=> = 4x + 4`
   
   `=>  = f(2x) - g(2x)`
   `=>  = (-6x + 7) - (4x + 4)`
   `=>  = -6x + 7 - 4x - 4`
   `=>  = -10x + 3`
6. `(g-f)(1/6)`
   
   `=> (g-f)(1/6) = g(1/6) - f(1/6)`
   
   For `g(1/6)`
   `=> g(1/6) = 2x + 4`
   `=> = 2(1/6) + 4`
   `=> = 1/3 + 4`
   `=> = 13/3`
   
   For `f(1/6)`
   `=> f(1/6) = -3x + 7`
   `=> = -3(1/6) + 7`
   `=> = -1/2 + 7`
   `=> = -13/2`
   
   `=>  = g(1/6) - f(1/6)`
   `=>  = 13/3 - (-13/2)`
   `=>  = 13/3 + 13/2`
   `=>  = (13(2) + 13(3))/6`
   `=>  = 65/6`