1) Consider the function: $f: \mathbb{R} \to \mathbb{R}$ (Real to Real Number), where $f(x)=2+x^2$, what would be all of the preimages of $3$?
1) $11$
2) $11$, $-11$
3) $1$, $-1$
4) $1$
2) Let $D = \{a,b,c\}$ and let $E = \{2,4\}$, we will define the function $f$ as $f: D \to E$ with the following facts.
$f(a)=2$
$f(b)=2$
$f(c)=4$
Based on this information, what is accurate regarding the function of $f$?
1) $f$ is a "bijection"
2) $f$ is considered to be "one-to-one"
3) $f$ is "onto" and "one-to-one"
4) $f$ is "onto"
The image $f(X) = \{ f(x) : x \in X \}$. So to find the preimage of $3$, you want $\{ x : f(x) = 3 \}$. So set $3 = 2 + x^{2}$. You should be able to take it from here.
For part (ii), a one-to-one function is a function such that $f(a) = f(b) \implies a = b$. Here, you have $f(a) = f(b)$, but $a \neq b$. An onto function is one such that $\forall{y} \in E$, $\exists{x} \in D$ such that $f(x) = y$. In other words, does every element in $E$ get mapped to? Clearly, yes.
A bijection is a one-to-one and onto function.