Function Notation and Evaluating

In this lesson you will: Read f(x) as \"the output of the rule f when the input is x\"; Substitute a number for every x in the rule, then simplify; Work backwards: given an output, solve for the input that produced it.

A name for the rule, and parentheses for the input

Writing y=3x-5 tells you how the output is built but gives the rule no name. Writing f(x)=3x-5 names it f, and now you can talk about a particular output without redrawing anything: f(4) is the output when the input is 4.

The letter f is a label, like a person's name. It never has a numerical value of its own. What has a value is f of something.

Substitute into every copy of the variable

To evaluate, replace each x in the rule by the input and simplify. Parentheses around the substituted value are worth writing every single time, especially for negatives: f(-2)=3(-2)-5 is unambiguous, while 3-2-5 is a different and wrong calculation.

If it helps, rewrite the rule with an empty box in place of the variable — f()=3-5 — and drop the input into every box.

Running the rule backwards

There are two questions you can ask, and they point in opposite directions. \"Find f(6)\" hands you the input and asks for the output — that is substitution. \"Solve f(x)=19\" hands you the output and asks for the input — that is Chapter 3 all over again.

For f(x)=2x+7, solving f(x)=19 means solving 2x+7=19, so 2x=12 and x=6. The function notation was only the wrapping.

Common mistake — Treating f(x) as f multiplied by x

This is the single most expensive misreading in the whole functions unit, and it hides for months because it usually still produces a number.

It looks like multiplication because that is what a letter next to a parenthesis has meant since Chapter 2: 3(x+1) really is 3 times (x+1). So a student meets f(x)=3x-5 and divides both sides by x to get f=3-(5)/(x), or reads f(4) as 4f.

But f is not a quantity. It is the name of a rule, and the parentheses are not multiplication parentheses — they hold the input. Nowhere on the page is f ever given a value of its own, and that absence is the tell.

The damage shows up later, when a student writes f(a+b)=f(a)+f(b) because that is how multiplication distributes. It is false for almost every function.

How to catch it: Test it with numbers. If f(x) were f times x, then f(2)+f(3) would equal f(5). Take f(x)=4x-1: f(2)=7 and f(3)=11, so f(2)+f(3)=18, while f(5)=19. They differ, so it is not multiplication. Run that check once and the idea sticks.