MHB Surjectivity of x²+3 for Real Numbers: Testing for Surjectivity

[markosheehan]

is the function x²+3 surjective for real numbers. how do you test for surjectivity in general?

 

[Jameson]

is the function x²+3 surjective for real numbers. how do you test for surjectivity in general?

Hi markosheehan,

Welcome to MHB! :)

Let's say a function $f$ maps some $X \in \mathbb{R}$ to $Y \in \mathbb{R}$. You might write that like $f: X \rightarrow Y|X,Y \in \mathbb{R}$ or maybe just $f:R \rightarrow R$. $f$ is surjective if every element in $Y$ is mapped to by at least one input in $X$. If $Y$ is the output of the function, we need to hit every value in $Y$ in order for $f$ to be surjective.

In this example we have $f(x)=x^2+3$. If $Y$ is all real numbers, can we output to all real numbers? Are there any numbers that we cannot map to?

In general the process of proving a function is subjective when going from $R \rightarrow R$ is to take some random $y \in Y$ and solve for it in terms of $x$. Is that possible here? What happens when you try to solve for $y$?

 

[Evgeny.Makarov]

Let's say a function $f$ maps some $X \in \mathbb{R}$ to $Y \in \mathbb{R}$. You might write that like $f: X \rightarrow Y|X,Y \in \mathbb{R}$ or maybe just $f:R \rightarrow R$.

This should say $X\subseteq\mathbb{R}$ instead of $X\in\mathbb{R}$. Also, $R$ in $f:R \rightarrow R$ should probably be $\mathbb{R}$, just like in the first occurrence.

 

[Jameson]

This should say $X\subseteq\mathbb{R}$ instead of $X\in\mathbb{R}$. Also, $R$ in $f:R \rightarrow R$ should probably be $\mathbb{R}$, just like in the first occurrence.

Good points, thank you! Proper notation is important.

Since the OP hasn't responded I will go ahead and answer the original question. $f(x)=x^2+3$ is not surjective for real numbers as it has a global minimum at $y=3$. Any number less than 3 is not mapped to by this function.