Two functions can be combined to make a new function through the basic arithmetic operations addition, subtraction, multiplication and division.

The following arithmetic rules show you what happens when you combine two functions f(x) and g(x).

**Sum**: (f + g)(x) = f(x) + g(x)**Difference**: (f – g)(x) = f(x) – g(x)**Product**: (f · g)(x) = f(x) · g(x)**Quotient**: (f / g)(x) = f(x) / g(x); note that the quotient f/g isn’t defined when g(x) = 0 because of division by zero.

The new functions are called an **arithmetic combination of functions.** The domain for the new function is all real number common to the domains of f(x) and g(x), with the exception of the quotient, where x cannot equal zero.

## Arithmetic Combinations of Functions: Examples

**Example question #1:** Find (f + g)(2) for the following two functions:

- f(x) = x
^{2}+ 3x – 7 - g(x) = 4x + 5 at x = 2.

**Solution**: (f + g)(2) = f(2) + g(2) = 3 + 13 = 16.

**Example question #2:** Find the quotient of the functions f(x) = x^{2} + 3x – 7 and g(x) = 4x + 5 at x = 3.

**Solution**: (f / g)(3) = f(3) / g(3) = 11 / 17.

**Example question #3:** Find (f · g)(x) for the following two functions:

- f(x) = 3x + 4
- g(x) = 2x
^{2}– 1.

**Solution**: We aren’t given an x-value here, so in order to combine these functions, substitute every value of x in f(x) = 3x + 4 with g(x) = 2x^{2} – 1:

3(2x^{2} – 1) + 4.

Expanding the expression gives the solution:

6x^{2} + 1.

## Graphical Solutions

If algebra isn’t your forte, you can use your graphing calculator to get a very good approximation for a difference of two functions. For example, let’s say you wanted to find the difference at x = 2 of

- f(x) = 2x + 1
- g(x) = x
^{2}+ 2x – 1

Algebraically, the rule is: (f – g)(x) = f(x) – g(x), which gives us:

(2(2) + 1) – (2^{2} + 2(4) – 1) = 5 – 7 = -2.

You can find the same solution on a graphing calculator by graphing three functions:

- f(x)
- g(x)
- f(x) – g(x)

I used Desmos.com to graph the functions; The solution at x = 2 (the green graph is #3 the difference function) is highlighted: