Common Algebra Mistakes
Example: Function Notation

The Goal

Evaluate f(x+h) for the function:

goal


The Mistake

Find the algebra mistake:

mistake

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


The Correction

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):

formula

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

formula

Replace x with 4 to find f(4):

formula

Note that 4 can be written as 3+1:

formula

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

formula

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

formula

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