Function notation
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Given f(x) = x^2 - 4x, find f(-2).
Common mix-up
In this example, a student wrote f(-2) = -4 + 8 = 4.
The student computed (-2)^2 as -4, then wrote f(-2) = -4 + 8 = 4.
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See why this works
What is f(-2) when f(x) = x^2 - 4x?
f(x) is a rule with a name, not multiplication. f(-2) means: feed -2 into the rule and see what comes out.
Replace every x in the rule with the number in the parentheses, then follow the order of operations.
Function notation is a substitution recipe. The rule f(x) = x^2 - 4x describes what to do with x: square it, then subtract four times the original. When the input is written inside the parentheses, as in f(-2), every x in the rule gets replaced by that input, and the parentheses keep the negative numbers together.
The dangerous moment is (-2)^2. The square applies to the whole quantity -2, not to just the 2, because of the parentheses. (-2)(-2) is positive 4. A student who writes -4 has squared the 2 and kept the negative sign hanging outside, and every later step inherits that wrong sign.
Continue through the rule. The second term of the rule is -4x, which becomes -4(-2) with the input inside parentheses. That is positive 8, because a negative times a negative is positive. The rule now reads f(-2) = 4 + 8 = 12, and the answer is 12.
One habit protects the sign mistakes: put parentheses around every substituted number. Then evaluate in the standard order: powers, multiplication, addition and subtraction. The earlier line, if written as -4 + 8 = 4, shows exactly where the squared negative got lost.
Keep this idea
f(a) means substitute a for every x inside the rule. Wrap each negative input in parentheses before evaluating.
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