A Basic Question about the Function 2*x^2

[The Student]

My textbook shows that when g(x) = 2*x^2 then g: [1, ∞) → [2, ∞), x ∈ ℝ. Why isn't the domain of x, [0, ∞)?

 

[pka]

My textbook shows that when g(x) = 2*x^2 then g: [1, ∞) → [2, ∞). Why isn't the domain of x [0, ∞)?

First let's get the notation straight.
Your function is \(\displaystyle g(x)={2^{{x^2}}}\).

Now you can say that \(\displaystyle g:[1,\infty)\mapsto [2,\infty)\). That is absolutely true.

BUT it is not the case that it implies that the domain of \(\displaystyle g\) is \(\displaystyle [1,\infty)\). It is not.

Actually the domain of \(\displaystyle g\) is all of \(\displaystyle \mathbb{R}~.\)

 

[The Student]

First let's get the notation straight.
Your function is \(\displaystyle g(x)={2^{{x^2}}}\).

Now you can say that \(\displaystyle g:[1,\infty)\mapsto [2,\infty)\). That is absolutely true.

BUT it is not the case that it implies that the domain of \(\displaystyle g\) is \(\displaystyle [1,\infty)\). It is not.

Actually the domain of \(\displaystyle g\) is all of \(\displaystyle \mathbb{R}~.\)

I am sorry, but I screwed up my question. The function that I wanted to ask about is 2 multiplied by the square root of x. Here is how I should have put it, 2*x^(1/2). I use * as a multiplication symbol. So the domain of g seems to be [0,∞) because x can equal 0; can't it? And then when x = 0 or > 0, then the range of g would seem to be the same as the domain, [0,∞). But they have g: [1, ∞) → [2, ∞). I just don't get it.

 

[JeffM]

I am sorry, but I screwed up my question. The function that I wanted to ask about is 2 multiplied by the square root of x. Here is how I should have put it, 2*x^(1/2). I use * as a multiplication symbol. So the domain of g seems to be [0,∞) because x can equal 0; can't it? And then when x = 0 or > 0, then the range of g would seem to be the same as the domain, [0,∞). But they have g: [1, ∞) → [2, ∞). I just don't get it.

A function may be defined over a specific domain.

So \(\displaystyle x \in [1, \infty) \implies g(x) = 2\sqrt{x}\) is a perfectly acceptable way to define g(x).

If that is how it is defined then the range is \(\displaystyle [2, \infty).\)

If no specific domain is implied, then it is implied that every real number for which the function generates a real value is included in the domain.

What EXACTLY does the problem ask?

 

[The Student]

A function may be defined over a specific domain.

So \(\displaystyle x \in [1, \infty) \implies g(x) = 2\sqrt{x}\) is a perfectly acceptable way to define g(x).

If that is how it is defined then the range is \(\displaystyle [2, \infty).\)

If no specific domain is implied, then it is implied that every real number for which the function generates a real value is included in the domain.

What EXACTLY does the problem ask?

It's just in my notes, "Definition: If a function f has domain A and range B, we write f : A → B". Then it shows "g(x) = 2√x, g : [1,∞) → [2,∞)". I am wondering why the domain of g is not [0, ∞). I realize that the notes aren't saying anything false, but all of the other examples show the whole domain and not just part of it.