How do I determine whether a function is onto or not without drawing the graphs?

Question

OK this is a general question. Please explain with examples if you can. How do I determine whether a function is onto or not without drawing the graphs? I know the function is onto when both the range and codomain are equal but in some cases I'm unable to calculate the range. So is there any other way to figure out what's onto and what's not? Please explain with examples!

Answer 1

"Onto" is essentially a representation of surjectivity. A function $f : A \rightarrow B$ is surjective if and only if

$$\forall y \in B. \exists x \in A. f(x) = y.$$

In words, it means that every element of $y$ can be mapped to by $f$ using some element in the domain of $f$, $A$. The way one can prove that a function is surjective is by taking an arbitrary element $y \in B$, and finding a specific element in $A$ (say, $x$) that when passed into $f$, produces $y$ (i.e. $f(x) = y$). To show that it is not surjective, find some element in $B$ that cannot be mapped to by $f$ using the domain $A$.

As one example, take the function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x) = 2x.$ Take $y \in \mathbb{R}.$ If we recognize by the properties of the reals that $\frac{y}{2} \in \mathbb{R},$ We see that $f\left(\frac{y}{2}\right) = 2\left(\frac{y}{2}\right) = y.$ Since $y \in \mathbb{R}$ was arbitrary, $f$ is indeed surjective.

However, take $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x) = x^2.$ If we choose a $y \in \mathbb{R}$ such that $y < 0,$ There is no real number $x \in \mathbb{R}$ such that $f(x) = y.$ Thus, $f$ here is not surjective.