Function Notation
Read, write, and evaluate functions using f(x) notation. Understand that f(x) is the output (not f times x), evaluate at numeric and algebraic inputs, and interpret f(a) = b statements in context.
A lesson of Algebra 1, in Functions & Relations, about 30 minutes.
By the end of this lesson, you will be able to read and write $f(x)$ fluently, evaluate a function at numbers and at expressions, say what $f(a) = b$ means in a real situation, and work out $f(g(x))$ at a value.
A Name for the Machine
You already know how to write a rule:
$$y = 3x + 5$$
Function notation writes the same rule with a name on it:
$$f(x) = 3x + 5$$
Read that as "f of x equals three x plus five". The $f$ is the machine's name, and the $x$ in brackets says what goes in.
$f(x)$ is not multiplication. It is not $f$ times $x$. It is the output of $f$ when you feed it $x$, and that is the single most common misreading of the whole topic.
What the name buys you is the ability to talk about ONE output without writing the rule out again:
$$f(4) \quad \text{means "the output when the input is 4"}$$
With $y = 3x + 5$ you have to say "the value of $y$ when $x$ is 4". With $f(4)$ you just point at it. That shortcut is the reason the notation exists, and it becomes essential the moment a page has two functions on it.
Reading It Out Loud
| Written | Said | Means |
|---|---|---|
| $f(4) = 17$ | "f of 4 is 17" | Input $4$ gives output $17$ |
| $f(a) = 0$ | "f of a is zero" | The input $a$ produces zero |
| $g(x) = x^2$ | "g of x equals x squared" | A second function, called $g$ |
| $f(g(2))$ | "f of g of 2" | Feed $2$ into $g$, then that answer into $f$ |
The letter is just a name. $f$ is conventional, $g$ and $h$ come next, and in science it is usually something meaningful: $C(t)$ for cost against time, $h(t)$ for height against time.
Example 1: Evaluating at a Number
Here is the whole method on one function, worked all the way through. $$f(x) = 3x + 5$$ Evaluating means substituting, exactly as it always did. The notation changes how you write it down, not what you do.
- Replace every x with 4, in brackets 3(4) + 5
- Work it out 17
- Write the finished statement, using the notation f(4) = 17
Worked Example 1: Your Turn
$$g(x) = x^2 - 2x$$
Feeding In Something That Is Not a Number
The input does not have to be a number. Whatever sits in the brackets goes wherever $x$ was.
$$f(x) = 3x + 5$$
$$f(a) = 3a + 5 \qquad f(2t) = 6t + 5 \qquad f(x + 1) = 3(x + 1) + 5 = 3x + 8$$
The last one is worth staring at. $f(x+1)$ does not mean "$f(x)$ then add one". It means the whole of $x+1$ goes into the machine, which is why it needs brackets and why the answer is $3x + 8$ rather than $3x + 6$.
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