Question

What does the term arbitrary mean?

Answer 1

when we say we take a arbitrary value we mean we take any random value in the domain of the function ...for example y = 1/x then arbitrary value of x means taking any value of x except 0 (as it is not in the domain)

Answer 2

Thank you. Can you explain me, what you mean by 0 not being in the domain??

Answer 3

well what i meant was that function y = 1/x is not defined at x = 0 as the denominator cannot be 0 (0 is not in the domain of this function) so if we say we take arbitrary value of x for this function we can take any random value except 0

Answer 4

According to dictionary.com, "arbitrary" when used in mathematics means "undetermined, not assigned a specific value."

I've come across the phrase in two different contexts. One is when determining the indefinite integral of a function. When you do this, you can only determine the integral to within an arbitrary constant, usually written C. Example: Integral(x) = x^2 + C.

The other time you find the phrase is in some proofs where you read "Take an arbitrary function of the form ...." It means pick a function, any function, of the form described.

The word arbitrary can cause some difficulty because it leads us to think of individual cases, when in fact it is used when dealing with broad groups of things. When we integrate the function f(x) to get F(x) + C, that also means that all functions of the form F(x) + C have f(x) as their derivative. When a proof works for any arbitrary function of a certain form, say linear functions, then that means that the proof holds for -all- linear functions.

I hope I haven't gone on too long or worse, confused you even more, but I'm new at this study group thing and trying to learn the ropes.

Answer 5

Thanks, that has helped a lot.