Function Notation: Understanding This Rule

function notationA function is a type of rule that operates with an input and then can produce an output. In essence, you have been using functions since you were in the first or second grade. Do you remember when your teacher would write ____ + 4 = 7, and you had to fill in the blank? You could hold up four fingers and keep adding fingers until you reached seven, or you could hold up seven fingers and lower enough fingers until you get to four. Either way, the answer, would always be three. These problems are seen at various levels of education and Udemy can help prepare you for basic or advanced functions in math class.

When you got to middle school, you still did problems like the one above, but you replaced the empty space with an x, as in x + 4 = 7, which is simple algebra. The reason they replaced boxes and spaces with letters is that it would be confusing when working harder math problems to have many different boxes and spaces that needed to be filled.

Function notation is part of algebra. You may have started out with y = 4x + 7; solve for y when x = -1. You would plug -1 into the x spot and solve like any other algebra problem. However, this can be confusing, so we now have function notation, or f(x). This reads as “f of x”, and is written as: f(x) = 4x + 7; f(x) = -1.

The Value of Functions 

However, function notation also gives you greater flexibility in formulas. You can have many different function notations and solve for them all easily. It helps so you do not get confused. For those looking for more than beginner information, Udemy has a wide range of advanced Algebra courses.

It is true that confusion is lessened, but there are a few rules that you must follow before function notation will work for you. In algebra, generally parentheses indicate that you must multiply; for example: (4+7)(8+8). With this example, you would add inside each set of parentheses and then multiply those two numbers together, giving you (11)(16)=176.

Now, however, parentheses do not necessarily mean multiplication when using f(x). This means to plug a value for the term x into a formula (f).

You must also understand that the x in f(x) means the argument of the function. In some cases, the problem will ask you what the argument is; you would simply answer whatever is in between the parentheses that stands for x.

Examples of Function Notation

An example of function notation is:

f(x) = x2 + 9x -1, find f(7).

f(7) = (7)2 + 9(7) – 1

= 49 + 63 – 1

= 111

As with many other things in life, there are different types of functions. One is called the piecewise function, as it is in pieces:



5x2 – 3, x < 1
X + 9, x ≥ 1

This function may look strange, because it is in two parts. The top part indicates to use that function if x is less than one, and the bottom part indicates that it should be used when x is greater to or equal to one. Which one you use depends on what your value of x is. If we are provided f(0), we will need to use the top function; if we are provided with f(1), we would use the bottom function

Using the above piecewise function, assume that you need to evaluate f(0). It would look like this:

f(0) = 5(0)2 – 3

= 0 – 3

= -3

Using the above function again, evaluate f(1). It would look like this:

f(1) = 1 + 9 = 10

Function notation isn’t just used in algebra; it is used in calculus and many other advanced math courses. It is taught at the algebra level so it will help you prep for advanced math, whether in high school or at college.

Function notation may not produce a numerical number, either. In some cases, the problem is meant to see if you can stay organized with your letters. An example is this:

If f(x) = 3x2 + 2x, find [f(x + p) – f(x)] / p.

Could this be any more confusing? It probably could, but it is answerable. First, you should break it up into pieces and find the expressions for f(x) and f(x + p). Then you would subtract, simplify, and divide.

We know that f(x) = 3x2 + 2x. We could write it another way, so that it looks like this:

f( ) = 3(  )2 + 2(  ). Into the parentheses, you would insert (x + p):

f(x + p) = 3(x + p)2 + 2(x + p). Simplify

= 3x2 + 6xp + 3p2 + 2x + 2p. We now need to solve for the brackets above:

f(x) = 3x2 + 2x

f(x + p) – f(x) (This is what is in brackets above and what we are solving for)

= [3x2 + 6xp + 3p2 + 2x + 2p] – [3x2 + 2x]

= 3x2 + 6xp + 3p2 + 2x + 2p – 3x2 – 2x. Group like terms together

= 3x2 – 3x2 + 6xp + 3p2 + 2x – 2x + 2p. Simplify.

= 6xp + 3p2 + 4x + 2p. Now take the entire problem from above and solve:

[f(x + p) – f(x)] / p

= 6xp + 3p2 + 4x + 2p / p

= p[6x + 3p + 2] / p

= 6x + 3p + 2. This is the answer to the problem.

Yes, the above looks odd, and there is no real way to get an actual number because we don’t know what the values of h and x are. However, looking through all of the process can show where organizational skills are necessary; it also shows that paying close attention to negatives is a must.

In some cases, you will be asked to “determine algebraically” if a function is odd or even. Of course, this can be easy to do on a graph, but algebraically means they want you to use algebra to determine odd or even.

When faced with a function, you must plug in the negative of x for f(x) and simplify. If the function you receive is the function you started with, it is even. If you end up with the opposite of what you started with, the function is odd. The answer can also be “neither even nor odd”.

More examples and explanations can be found in the Udemy Algebra and Calculus courses. If you are comfortable and ready to move onto the next level, Udemy offers a great Intermediate Algebra course that can help to enhance and further the skills that you already have.