 ## How to Add and Subtract Functions

### What does "Function Notation" mean?

The function notation is a quick and easy way to write functions, making it easy to write, understand, and evaluate them. Algebraic expressions are used in function notation, to sum up how a function changes the input value to get the desired output value. One of the most common ways to write a function in math is $$f(x)$$, which is read as "$$f$$ of $$x$$". The relationship between the input and output values is shown by the function $$f(x)$$.

The sum, difference, product, and quotient are only defined for real functions based on their shared domain. Even if their domains are the same, these operations don't make sense for general functions because the sum, difference, product, and quotient may or may not make sense for the elements in their common domain.

### How to Add and Subtract Functions Step by Step

• We can add and subtract functions just like we can add and subtract numbers. For instance, if we had two functions, $$f(x)$$ and $$g(x)$$, we could make two new functions: $$(f \ + \ g)(x)$$ and $$(f \ - \ g)(x)$$.
• When a function has polynomials, combine the like terms to add or subtract functions.
• When evaluating new functions, you should substitute the input value to find the function's value.

### Example

If $$f(x) \ = \ 2x^2 \ + \ 5x \ - \ 7$$ and $$g(x) \ = \ 5x^2 \ - \ 6x \ + \ 2$$, find: $$(f \ + \ g)(x)$$, $$(f \ - \ g)(x)$$

Solution:

$$(f \ + \ g)(x) \ = \ (2x^2 \ + \ 5x \ - \ 7) \ + \ (5x^2 \ - \ 6x \ + \ 2) \ = \ 7x^2 \ - \ x \ - \ 5$$

$$(f \ - \ g)(x) \ = \ (2x^2 \ + \ 5x \ - \ 7) \ - \ (5x^2 \ - \ 6x \ + \ 2) \ = \ -3x^2 \ + \ 11x \ - \ 9$$

### Exercises for Adding Subtracting Functions

1) $$h(x) = -5x \ + \ 4 \\ g(x) = 6x \ + \ 2$$
Find $$(h + g)(x)$$

2) $$h(x) = 3x \ + \ 3 \\ g(x) = 4x \ + \ 1$$
Find $$(h + g)(x)$$

3) $$h(x) = 10x \ + \ 5 \\ g(x) = 4x \ + \ 4$$
Find $$(h + g)(x)$$

4) $$h(x) = -7x \ + \ 7 \\ g(x) = 8x \ + \ 2$$
Find $$(h - g)(x)$$

5) $$h(x) = -2x \ + \ 8 \\ g(x) = 7x \ + \ 4$$
Find $$(h - g)(x)$$

6) $$h(x) = -10x \ + \ 6 \\ g(x) = 3x \ + \ 4$$
Find $$(h - g)(x)$$

7) $$h(x) = -3x \ + \ 7 \\ g(x) = 4x \ + \ 1$$
Find $$(h - g)(x)$$

8) $$h(x) = 11x \ + \ 7 \\ g(x) = 7x \ + \ 4$$
Find $$(h + g)(x)$$

9) $$h(x) = -6x \ + \ 5 \\ g(x) = 2x \ + \ 4$$
Find $$(h + g)(x)$$

10) $$h(x) = 4x \ + \ 8 \\ g(x) = 2x \ + \ 3$$
Find $$(h + g)(x)$$

1) $$h(x) = -5x \ + \ 4 \\ g(x) = 6x \ + \ 2$$
Find $$(h + g)(x)$$$$\\ \color{red}{(h + g)(x) = (-5x \ + \ 4) + (6x \ + \ 2) \\ (h + g)(x) = 1x \ + \ 6}$$
Solution
Step 1:
Write out the two functions you want to add or subtract. For example you are asked to add functions $$f(x) = 2x + 4$$  and  $$g(x) = 3x - 1$$. In this step, you must add write them together: $$-5x + 4 + 6x + 2$$
Step 2: Now you need to add or subtract the variables and numbers: $$-5x + 6x = 1x$$  and  $$4 + 2 = 6$$
Step 3: Write a new function and write the variables and numbers you get in step 2: $$p(x) = 1x + 6$$
2) $$h(x) = 3x \ + \ 3 \\ g(x) = 4x \ + \ 1$$
Find $$(h + g)(x)$$$$\\ \color{red}{(h + g)(x) = (3x \ + \ 3) + (4x \ + \ 1) \\ (h + g)(x) = 7x \ + \ 4}$$
Solution
Step 1:
Write out the two functions you want to add or subtract. For example you are asked to add functions $$f(x) = 2x + 4$$  and  $$g(x) = 3x - 1$$. In this step, you must add write them together: $$3x + 3 + 4x + 1$$
Step 2: Now you need to add or subtract the variables and numbers: $$3x + 4x = 7x$$  and  $$3 + 1 = 4$$
Step 3: Write a new function and write the variables and numbers you get in step 2: $$p(x) = 7x + 4$$
3) $$h(x) = 10x \ + \ 5 \\ g(x) = 4x \ + \ 4$$
Find $$(h + g)(x)$$$$\\ \color{red}{(h + g)(x) = (10x \ + \ 5) + (4x \ + \ 4) \\ (h + g)(x) = 14x \ + \ 9}$$
4) $$h(x) = -7x \ + \ 7 \\ g(x) = 8x \ + \ 2$$
Find $$(h - g)(x)$$$$\\ \color{red}{(h - g)(x) = (-7x \ + \ 7) - (8x \ + \ 2) \\ (h - g)(x) = -15x \ + \ 5}$$
5) $$h(x) = -2x \ + \ 8 \\ g(x) = 7x \ + \ 4$$
Find $$(h - g)(x)$$$$\\ \color{red}{(h - g)(x) = (-2x \ + \ 8) - (7x \ + \ 4) \\ (h - g)(x) = -9x \ + \ 4}$$
6) $$h(x) = -10x \ + \ 6 \\ g(x) = 3x \ + \ 4$$
Find $$(h - g)(x)$$$$\\ \color{red}{(h - g)(x) = (-10x \ + \ 6) - (3x \ + \ 4) \\ (h - g)(x) = -13x \ + \ 2}$$
7) $$h(x) = -3x \ + \ 7 \\ g(x) = 4x \ + \ 1$$
Find $$(h - g)(x)$$$$\\ \color{red}{(h - g)(x) = (-3x \ + \ 7) - (4x \ + \ 1) \\ (h - g)(x) = -7x \ + \ 6}$$
8) $$h(x) = 11x \ + \ 7 \\ g(x) = 7x \ + \ 4$$
Find $$(h + g)(x)$$$$\\ \color{red}{(h + g)(x) = (11x \ + \ 7) + (7x \ + \ 4) \\ (h + g)(x) = 18x \ + \ 11}$$
9) $$h(x) = -6x \ + \ 5 \\ g(x) = 2x \ + \ 4$$
Find $$(h + g)(x)$$$$\\ \color{red}{(h + g)(x) = (-6x \ + \ 5) + (2x \ + \ 4) \\ (h + g)(x) = -4x \ + \ 9}$$
10) $$h(x) = 4x \ + \ 8 \\ g(x) = 2x \ + \ 3$$
Find $$(h + g)(x)$$$$\\ \color{red}{(h + g)(x) = (4x \ + \ 8) + (2x \ + \ 3) \\ (h + g)(x) = 6x \ + \ 11}$$

## Adding and Subtracting Functions Practice Quiz

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