Interval functions

Question

I'm kinda a greenhorn in maths and young and unexperienced, but one thought popped in my head that google couldn't satisfy.

Let $g(x) = \sin(x)$.

Then wouldn't be $g([a,b]) = \{ (x, y): a \leq x \leq b \; \text{and} \; y = g(x) \}$ pretty fine?

Is $g([a,b])$ meaningless or incorrect? If it's not, what would be the result? Is there a function that argument is an interval? Or this whole idea is based on my incorrect vision of maths skewed by programming?

I'm sorry for unskilful use of MathJax and maths at all, it's just pure curiosity.

and $g([a,b])={g(x): x\in [a,b]}$ I find the easiest expression to understand what does $g([a,b])$ means — Joaquin Liniado, Jun 08 '15 at 22:48
I meant like a set of pairs $(x,y)$ where $x$ is an every value from range $[a,b]$ and $y$ is a corresponding $y = g(x)$. — Are, Jun 08 '15 at 22:51

score 3 · Accepted Answer · answered Jun 08 '15 at 22:50

3

The set $$ \{ (x, y): a \leq x \leq b \; \text{and} \; y = g(x) \}$$ is commonly called the graph of the function $g(x)$ on the interval $[a,b]$.

It is common to use the notation $g(S)$, for set $S$, to mean the set of all values $g(x)$ takes on over the set $S$, i.e. $$\{g(x):x\in S\}$$

answered Jun 08 '15 at 22:50

vadim123

82,796

Absolutely, $[a,b]={x \in \mathbb{R} : a \leq x \leq b}$ in set notation. – Joaquin Liniado Jun 08 '15 at 23:01

Interval functions

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