Some problems provide the opportunity for more than one mistake.

The Goal

Evaluate f(x+h) for the function:

The Mistakes

Find the mistakes:

1.

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

2.

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

A Correct Solution

Explanations

In the first mistake f(x+h) is computed to be f(x)+h. 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.

In the second example the +h is again in the wrong place.

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 0.5 to find f(0.5):

Note that 0.5 can be written as 0.4+0.1:

Following the pattern we can find f(0.4+h):

And f(x+h) is a matter of replacing the 0.4 by x: