The Goal

Evaluate f(x+h) for the function:

The Mistake

Find the algebra mistake:

Need a hint? Look carefully at the red part of the algebra:

The Correction

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 5 to find f(5):

Note that 5 can be written as 4+1:

Following the pattern we can find f(4+h):

And f(x+h) is a matter of replacing the 4 by x: