Is f(x) = √9x a bijection?

[dirtypurp]

Let f : N −→ R and f(x) = √ 9x

The domain is all natural numbers: {0, 1, 2, 3, ...}

The codomain is all real numbers.

The range i believe is [0, +infinity)

I believe that although the above is a function since every input of x provides a output that fits in our codomain. I also believe that this is a injective function (one to one correspondence) since f(x)=f(y); x=y. However I do not believe that it is a bijection since not every output we get, which is considered to be our range, is equal to the codomain which is all real numbers. For example no negative output can be given.

Does anyone care to agree or disagree?

 

[Infrared]

The range is not $[0,\infty)$. For example, $1$ is not in the range.

 

[dirtypurp]

The range is not $[0,\infty)$. For example, $1$ is not in the range.

So what is the range? and if so the function isn't bijection then based on what you said

 

[Infrared]

The range is the set of real numbers of the form $\sqrt{9n}$ (or equivalently $3\sqrt{n}$), where $n$ is a natural number. There's not a simpler way of writing it.

As you noted, the function is not a bijection because it is not surjective.

 

[dirtypurp]

The range is the set of real numbers of the form $\sqrt{9n}$ (or equivalently $3\sqrt{n}$), where $n$ is a natural number. There's not a simpler way of writing it.

As you noted, the function is not a bijection because it is not surjective.

ah okay that makes sense however the function is injective since every output is only given by one input.

thank you for your help

 

[zinq]

I think it's important to mention that the original question is stated in a totally unclear fashion, because it could be that the square root applies to 9x, but it also could be that the square root applies only to 9 before the result (3) is multiplied by x.

The symbol √9x should never be used without parentheses, as either (√9)x or √(9x).

 

[pbuk]

The question in the OP is clear to me (and I think the others who have provided substantive answers), although it would of course be even clearer using LaTeX: $f(x) = \sqrt{9x}$. The alternative interpretation is that the question relates to $f(x) = \sqrt{9}x = 3x$, which would be nonsensical. And before you say 'but surely it would be better to write f(x) = 3√x', consider that this could easily be confused with $f(x) = \sqrt[3]{x}$.

 

[zinq]

Not at all nonsensical! Just probably too simple to ask about. Which is different.

One's text should not depend on others' decisions about what is probably too simple to ask about.