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`
- `(g+f)(x)`
`=> (g+f)(x) = g(x) + f(x)`
`=> = (2x + 4) + (-3x + 7)`
`=> = 2x - 3x + 4 + 7`
`=> = -x + 11` - `(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` - `(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` - `(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` - `(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` - `(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`
