How Does Function Notation Work?

In calculus, we will need to  take a function (x) and write out (x+h) for that function. Let’s look at how to do this properly.

To do a problem like this, you need to understand exactly what the x in (x) represents and what the f represents. Let’s look at the function (x) = x2x. A function is a process. In this case, it is the process of

1. Square the input
2. Take the result and subtract the input

Notice that there is no mention of the x in the formula. That is because it is a placeholder representing the input. There is nothing special about x. We could have just as easily used a different letter as a placeholder for the input. If I had wanted to call the input t, I would have written

(t) = t2 – t

If the input had been represented by the word dog, I would have written

(dog) = dog2 – dog

The input variable is simply a placeholder…if a number is put in its place like 7, we get

(7) = 72 – 7 = 42

Notice that the process is the same. Square the input and subtract the input from the result. In this case, the input is 7 so we are squaring 7 and then subtracting 7 from the result.

Many students are confused by f(x+h). Now the input is represented by x+h instead of x. This means we need to square it and then subtract x+h from the result.

(x+h) = (x+h)2 – (x+h)

We can simplify this by foiling out the square,

(x+h) = (x+h)(x+h) = x2 +2xh + h2

And removing the parentheses after the subtraction we get

(x+h) =  x2 +2xh + h2xh

The handout below has more examples with this function.