Function Notation and Evaluating
f(x) names a rule, not a multiplication: it tells you what happens to whatever gets fed into the machine. To evaluate, replace every x with the input and simplify.
Think of f as the name of a machine, and x as a placeholder for whatever you drop in. f(x) = 2x2 - 5x + 1 says: take the input, square it and double that, then subtract five times the input, then add 1. The parentheses don't mean multiply f by x. They mean apply f to x. That's why f(3) isn't f times 3 - it's the output you get after running 3 through the rule.
This matters most when the input isn't a bare number. If you feed in a whole expression, like 2a+1, every single x in the rule gets replaced by that entire expression, parentheses and all. So f(2a+1) means you square (2a+1), not just a, and you subtract 5 times (2a+1), not just 5 times a. The machine doesn't care how complicated the input looks - it treats the whole thing as one object going in.
This is also exactly why f(x+2) and f(x)+2 split apart. f(x+2) sends x+2 through the machine: everything gets squared or multiplied together as a unit. f(x)+2 runs x through the machine first, gets some output, and only then tacks on 2 at the end. Same machine, two different moments for the +2 to happen. This idea is the foundation for composing functions, like g(h(2)), where you evaluate the inside function first and feed that whole result into the outside one.
Key terms
- Function notation
- A way of writing a rule, like f(x), where the letter names the rule and x marks the input.
- Evaluate
- To find the output of a function by substituting a specific value or expression for every x in the rule.
- Input
- Whatever is placed inside the parentheses, which gets substituted for x throughout the rule.
- Composition
- Feeding the output of one function into another, written g(h(x)), evaluated inside out.
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End of topic.