List of top Mathematics Questions on Types of Functions

Check whether the function \[ f:\mathbb{R}\setminus\{3\}\rightarrow\mathbb{R} \] defined by \[ f(x)=\frac{x-2}{x-3} \] is onto (surjective) or not.

Show that $f : \mathbf{R}_+ \to [-5, \infty)$ given by $f(x) = 4x^2 + 4x - 5$ is both one-one and onto where $\mathbf{R}_+ = [0, \infty)$. Also, find $p \in \mathbf{R}_+$ such that $f(p) = 3$.

Check whether the function \( f : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z} \) defined as \( f(x, y) = (2y, 3x) \) is injective (one-to-one) or not.

\(\text{ If the function } f : \mathbb{N} \to \mathbb{N} \text{ is defined as } f(n) = \begin{cases} n - 1, & \text{if } n \text{ is even} \\ n + 1, & \text{if } n \text{ is odd} \end{cases} \text{, then:}\)
(A) f is injective
(B) f is into
(C) f is surjective
(D) f is invertible
Choose the correct answer from the options given below: