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A function that has range as set of real numbers

OR

A function which has either ${R}$ or one of its subsets as its range.

So , in 2nd definition

Does the 2nd statement mean that the functions range is either a real number or a subset of real numbers ?

Could you give an example for 2nd definition.Especially in subset case

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  • $\begingroup$ Those definitions are both the same. The range is a (sub)set of $\mathbb R$. $f(x) = \sin x$ has $[-1,1]\subset \mathbb R$ as it's range. $g(x) = x^2$ has $[0,\infty)\subset \mathbb R$ as its range and $h(x) =x^3$ has $\mathbb R\subset \mathbb R$ as its range. $\endgroup$
    – fleablood
    Commented Jan 16, 2021 at 7:20

2 Answers 2

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A real function is a function whose codomain is $\Bbb R$. So, its range is necessarily a subset of $\Bbb R$, but it doesn't have to be the whole $\Bbb R$.

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  • $\begingroup$ Thank you very much $\endgroup$
    – Srijan
    Commented Jan 16, 2021 at 7:30
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As $\mathbb R$ is a subset of itself and is a set of real numbers both those definitions are the same.

If $f(x) = \sin x$ has $[-1,1]\subset \mathbb R$ as it's range.

$g(x) = x^2$ has $[0,\infty)\subset \mathbb R$ as its range

and $h(x) =x^3$ has $\mathbb R\subset \mathbb R$ as its range.

And $j(x) = 7$ has $\{7\} \subset \mathbb R$ as its range.

All are real functions.

But as Jose Carlos Santos points out the actual range doesn't matter. So long as it is specified that the function's codomain (a set of which all the values of the function are mapped to) is the real numbers you don't actually have to specify the exact range.

In all my above examples it's enough to say they all map to the real numbers.

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  • $\begingroup$ Thank you very much $\endgroup$
    – Srijan
    Commented Jan 16, 2021 at 7:29

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