If a real-valued function $f:[a, \infty) \rightarrow [b, \infty)$ defined by $f(x) = 2x^2 - 3x + 5$ is a bijection,then $3a + 2b =$

Consider the sets $A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$,$B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$,$C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \leq 4\}$,and $D = A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:

If $f : R \to R$ is defined by $f(x) = 2x + \cos x$,then $f$ is

Check the injectivity and surjectivity of the following function $f : N \rightarrow N$ given by $f(x) = x^{3}$.

Given below are two statements:
Statement $I$: The function $f:R \rightarrow R$ defined by $f(x) = \frac{x}{1+|x|}$ is one-one.
Statement $II$: The function $f:R \rightarrow R$ defined by $f(x) = \frac{x^{2}+4x-30}{x^{2}-8x+18}$ is many-one.
In the light of the above statements, choose the correct answer from the options given below:

Let $f: R \rightarrow R$ be defined by $f(x) = x^{2} - \frac{x^{2}}{1+x^{2}}$ for all $x \in R$. Then,