Combining Functions

In Algebraic calculations, you would most deal with the systems of Linear Equations that can also be correlated with finding the Combination of two Functions. Combining functions can Mean several possible operations like adding up two functions together, subtracting one function from the other, multiplying one function by the other, dividing one function by the other, or substituting one function into the other.
To perform combining of functions arrange them such that similar terms are present in the same order. For instance, let us consider two functions given as:
 F (x) = x² - 4x + 5 and
G (x) = 4x + 4 – y.
Here y represents the function G (x). So, we need to rearrange the second equation by adding “y” both sides to obtain 2G (x) = 4x + 4, then divide both sides of the equation by 2 to obtain G (x) = 2x + 2.
Now let us perform the combining operations on the functions F (x) and G (x). First we go for addition where the like terms of the function G(x) are added to those of the function F (x). We will add 2x to - 4x to obtain -2x, and you will add 2 to +5 to obtain 7. Thus,
F (x) + G (x) = x² - 2x + 5
Similarly,
F (x) - G (x) = x² - 4x + 5 – (2x + 2) = x² - 6x + 3
F (x) / G (x) = x² - 4x + 5 * (2x + 2) = 2x³ – 8x² + 10x + 2x² – 8x + 10 = 2x³ – 6x² + 2x + 10
F (G (x)) = (2x + 2) ² – 4 (2x + 2) + 5 = 4x² + 4 + 8x – 8x – 8 + 5 = 4x² +1