Definition: Function notation writes the output of a function f for an input x as f(x), read as f of x, which names the rule and shows the input at once.
What does f(x) mean?
The symbol f(x) is not f times x. It is the output the function f gives when the input is x. The letter before the parentheses names the rule, so you can handle several functions in one problem, such as f, g, and h.
How do you evaluate it?
Replace every x in the formula with the number or expression inside the parentheses, then simplify. Use parentheses around substituted negatives to protect the signs.
Example. Let f(x) = 2x2 – 5x + 1. Then f(-3) = 2(-3)2 – 5(-3) + 1 = 18 + 15 + 1 = 34. And f(a + 1) = 2(a + 1)2 – 5(a + 1) + 1 = 2a2 – a – 2.
Which errors are common?
Students often square only the number and forget the negative sign: (-3)2 is 9, not -9. Another error is treating f(a + 1) as f(a) + 1; these are different in general. Finally, f(x) = 10 asks you to solve for x, not to evaluate.