Define a/the function to be

Question

Which is the correct way to define a function?

1. Define a function f(x) to be f(x)=2x
2. Define the function f(x) to be f(x)=2x.

Since f(x)=2x is a specific function, my guess is "the" is correct. But that specification is determined after saying "to be", so it might be "a function".

Also, if there are any other expression to define a function, I would like to know.

Any help is appreciated.

Answer 1

Is "function f(x)" focus of your writing/speech up to this point? - they would not be if you are listing some steps that functions are involved in to accomplish something else.

• If not, don't use the article.

If they are ...

• Should the reader already know which "function f(x)" you are talking about? You may not have defined the function until now, but perhaps you said in previous text/speech "We will need a function that does X / we will need a function for purpose X" or something similar. So the would indicate you are talking about that same function, just now getting around to defining it.

• If it does not matter which "function f(X)", then use a/an.

Most likely the article is not needed.

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Since f(x)=2x is a specific function, my guess is "the" is correct.

"Specific" in regards to articles means "a particular X - that we have talked about before or physically experienced together - out of all possible X's"

It's specific to you, but it's not "specific" yet to your audience if you haven't talked to the audience about it before or if it's not a shared experience, e.g. not on a chalkboard/whiteboard or introductory image they should have seen.

Answer 2

In math, I've usually seen "the" when defining symbols:

Define the function f(x) to be f(x) = 2x.

However, that sentence is a bit redundant, and I'd phrase it like

Define the function f(x) = 2x.

There are other ways to do it; below are a couple of possibilities. You'll see others in mathematical textbooks and manuscripts.

• Define f(x) = 2x. (when it's clear that f is a function)
• Let the function f(x) be defined by f(x) = 2x.

Answer 3

(1) A function f is defined as f(x)=2x

We define a function f by f(x)=2x

We define a function f to be f(x)=2x

are possible. Note that function f is defined on real line (=space) containing x. That is, functions on real line are many so that a is correct.

(2) For a variable x, we define the function f(x) by f(x)=2x

For a function f, we define the operator L(f) by L(f)=2f.

are possible. Here we use the because before construction of f we introduce some.