The Goal

Evaluate f(x+h) for the function:

The Mistake

Find the algebra mistake:

(Roll the mouse over the algebra to see a hint in red)

The Correction

(Roll the mouse over the area above to see the correction in blue)

An Explanation

There are very few functions f(x) for which the equation f(x+h) = f(x) + h is true (it is true if y = f(x) is a straight line with slope 1).
The mistake in thinking that f(x+h) = f(x) + h is true in general comes from not properly understanding function notation.
In f(x) the x is a placeholder. To evaluate f(x+h) we must put x+h in for that placeholder precisely where x appears in f(x) - no more and no less.
Follow the sequence of examples:

The function f(x):

Whatever we replace x with replaces x in the formula (even something silly like @):

Replace x with 4 to find f(4):

Note that 4 can be written as 3+1:

Following the pattern we can find f(3+h):

And f(x+h) is a matter of replacing the 3 by x: