In calculus, we will need to take a function f (x) and write out f (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 f (x) represents and what the f represents. Let’s look at the function f (x) = x2 – x. A function is a process. In this case, it is the process of
- Square the input
- 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
f (t) = t2 – t
If the input had been represented by the word dog, I would have written
f (dog) = dog2 – dog
The input variable is simply a placeholder…if a number is put in its place like 7, we get
f (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.
f (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
f (x+h) = x2 +2xh + h2 – x – h
The handout below has more examples with this function.