Just like you can add, subtract, multiply, or divide numbers, you can do those same operations with functions.
Suppose you have two functions
To add these two functions, we have two ways to write the notation. We can write addition as which is how most math books indicate addition or which is what most students find most useful. To find we only need to add the two functions together. Subtraction
The parentheses that were included in the previous step were not absolutely necessary. However, it can be helpful to use parentheses to separate the functions. It is also helpful to use this notation when subtracting and multiplying functions.
Now we need to combine like terms (link to basic operations-simplifying.doc)
So we have
As with addition, we can write the notation for subtraction of two functions as or . However, with addition it did not matter if we wrote or . The answer is the same in both cases. You might want to verify this for yourself. But with subtraction, the order definitely matters. Think about it this way. When you are adding two numbers, you can put them in any order, but the same is NOT true when subtracting two numbers. The same holds true when working with functions. So if we want to find , we must first write down and then write down and then perform the subtraction. Multiplication
You should be able to see how helpful the parentheses are in this problem, since we will have to change the sign of every term in the second set of parentheses. At this point, we need to combine like terms to get our answer.
So we have
As with addition, the order that multiplication is listed in will not affect the final answer. And as with subtraction, the use of parentheses will be important to us. Division
To find we will simply multiply our two functions.
To simplify, we must make use of the distributive property. Remember the distributive property says that we have to multiply every term in the first set of parentheses by every term in the second set of parentheses.
After using the distributive property, we will simplify and combine like terms.
In some ways, division of two functions can be the easiest of the four operations. To find we only have to create a rational function by putting in the numerator and in the denominator. This gives us
Notice that when working with addition, subtraction, and multiplication, we did not worry about the domain of our newly created functions. However, with division when we create a rational function, we have to be concerned about having zero in the denominator.
Let’s call our newly created function . So we have . The domain of is all real numbers except x = -3. If you need to review the domain of a function, click here.
You can see that combining functions is not quite as easy as combining two numbers, but the process is very similar. There is another way to combine two functions which is called composition of functions. To learn more, click here. (link to functions-composition.doc) Evaluation
In each of the operations we looked at, we found a new function and that was the end of our problem. It is possible to be asked to evaluate this newly created function at a particular value. If you need to review how to evaluate a function, click here. (link to functions-evalution.doc)
Let’s go back and re-visit our addition problem. We had and and found . Suppose we were also asked about . It is the same process used for evaluating any other function. We need to substitute the value of 3 into the new function everywhere we see an x. Doing so will give us .
You should substitute the value of 3 into both the f function and the g function and then add those values to make sure that also gives you 19. This shows that it is possible to evaluate each function individually and then combine the two values. However, it is usually a more expedient method to combine the two functions and then do the evaluation. But it does provide a good way for you to check your work.