Answers · The mathematics underneath
Is f(x) just f times x, or is it something else?
In shortNo. f(x) means the rule named f applied to x, giving one output value; it is not f multiplied by x.
It is something else. The letter f names a rule, and f(x) is the single value that rule returns when it is fed x. The bracket is not a multiplication sign; it is an inlet. So f(x) is read as f of x, not f times x, and the whole expression is a quantity once the rule has run.
The distinction is easiest to hold by naming the parts. In f(x), the letter before the bracket is the name of the machine, the material inside the bracket is what is being handed to it, and the whole expression names what comes out. That output is a number, or a vector, or whatever kind of object the rule produces, and it can then be added, compared, squared or fed into another rule like any other value. The rule itself is f, without brackets. It is a different kind of object: something that can accept an input rather than something that has already accepted one.
Take the rule f(x)=2x+1. Feeding in 3 means replacing every x on the right by 3: twice 3 is 6, adding 1 gives 7. That 7 is f(3). It is already a value, so f(3)+1 is 7+1=8. Nothing in this calculation multiplies f by 3. The letter f never stands for a number here; it stands for the operation that turns an input into twice that input plus 1. If you wanted multiplication by a number, that number would have to be written, or declared, as a number.
Why the multiplication guess fails
The multiplication reading fails because there is no numeric value for f in the line. In f(x)=2x+1 the symbol f is being defined, not measured. It is the name of the rule, so f times x would require a number f, and none has been supplied. The same arrangement of marks can mean multiplication, but only when the letter before the bracket has been introduced as an ordinary number. If w=5, then w(x+1) is multiplication: with x=3, the bracket gives 4, and five times 4 is 20. With the rule f above, f(x+1) is not 20; it is twice 4 plus 1, which is 9. The marks are nearly identical, and the declaration is what tells them apart.
That is the uncomfortable part of the notation: f(x) and w(x) look alike, and only the earlier declaration separates a rule from a number. A reader who has not found the declaration has to guess, and guessing wrong turns an output into a product. The safe habit is to ask what the name before the bracket was introduced as. If it was introduced as a function, the bracket means fed with. If it was introduced as a number, the bracket means multiplication. A subscript or a semicolon can add further information about which inputs are held still, but it does not change the basic reading of the bracket.
The machine and its output
The difference between f and f(x) is the difference between a rule and a result. Sentences about f are about the machine: f is smooth, f has a minimum, f is increasing. Sentences about f(x) are about a value: f(3) is 7, f(3)+1 is 8. The test is whether the expression could still be fed something. f can be fed an input, so it is the rule. f(3) cannot be fed an input without changing the expression, so it is a result. This is also why f(g(x)) makes sense: g(x) is a value, and that value becomes the input to f.
Where the reading changes
The reading changes only when the symbol changes role. If f is declared to be a number, then f(x) really is multiplication, and no amount of looking at the bracket alone will reveal that. The notation also permits a common abuse: people write the function f(x) when they mean the rule f. In that phrase f(x) is being used as a name for the machine, not as one output. Read it as shorthand, not as a contradiction. Outside those two cases, f(x) is an application: name outside, input inside, output as a whole.
A final check is to read the line aloud. f of x names an output. f times x names a product, and a product needs two numbers. If one of the things you are about to multiply is a rule rather than a number, the multiplication reading has already gone wrong. Once the output exists, it behaves like any other quantity: it can be added to something, compared with f(y), or passed into another rule. What it cannot do is turn back into f, because a value is not a machine.
Worth keeping
- f is the rule and f(x) is one value that rule returns when fed x.
- The same bracket can mean multiplication, but only when the name before it was declared as a number.
- Read nested brackets from the inside out, naming each intermediate result.
Where this goes next
This question is one step of Reading the Notation, a course that works through its subject from the beginning. The page that teaches this part of it is Something In, Something Out, and it reads the whole way through whether or not you have an account.
What an account changes is the practice: problems standing under every idea, marked with a reason rather than a tick, in a copy of the course that is yours.
Read that pageThe whole courseOther questions people ask about this
- Why does any number to the power zero equal one?Any non-zero number to the power zero is one because no copies are multiplied, and an empty product is one, not zero.
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- Does a subscript like a_ij mean a times i times j?No, a subscript marks which row and column an entry belongs to rather than multiplying by the letters underneath.
- What is the difference between variance and standard deviation?They measure the same spread: variance is the average squared distance from the mean, standard deviation is its square root, so you report the standard deviation and calculate with the variance.
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