Let $f : R \rightarrow R$ be a function such that $f ( x ) = \frac { x ^ { 2 } + 2 x + 1 } { x ^ { 2 } + 1 }$. Then
(1) $f ( x )$ is many-one in $( - \infty , - 1 )$
(2) $f ( x )$ is many-one in $( 1 , \infty )$
(3) $f ( x )$ is one-one in $[ 1 , \infty )$ but not in $( - \infty , \infty )$
(4) $f ( x )$ is one-one in $( - \infty , \infty )$

The function $f : \mathbb{R} \to \mathbb{R}$, $f ( x ) = \frac { x ^ { 2 } + 2 x - 15 } { x ^ { 2 } - 4 x + 9 } , x \in \mathbb { R }$ is
(1) one-one but not onto.
(2) both one-one and onto.
(3) onto but not one-one.
(4) neither one-one nor onto.

Let $f , g : \mathbf { R } \rightarrow \mathbf { R }$ be defined as : $f ( x ) = | x - 1 |$ and $g ( x ) = \begin{cases} \mathrm { e } ^ { x } , & x \geq 0 \\ x + 1 , & x \leq 0 \end{cases}$ Then the function $f ( g ( x ) )$ is
(1) neither one-one nor onto.
(2) one-one but not onto.
(3) onto but not one-one.
(4) both one-one and onto.

Let $f ( x ) = \left\{ \begin{array} { c c c } - \mathrm { a } & \text { if } & - \mathrm { a } \leq x \leq 0 \\ x + \mathrm { a } & \text { if } & 0 < x \leq \mathrm { a } \end{array} \right.$ where $\mathrm { a } > 0$ and $\mathrm { g } ( x ) = ( f ( x \mid ) - | f ( x ) | ) / 2$. Then the function $g : [ - a , a ] \rightarrow [ - a , a ]$ is (1) neither one-one nor onto. (2) onto. (3) both one-one and onto. (4) one-one.

The function $f : (-\infty, \infty) \rightarrow (-\infty, 1)$, defined by $f(x) = \frac{2^{x} - 2^{-x}}{2^{x} + 2^{-x}}$ is:
(1) Neither one-one nor onto
(2) Onto but not one-one
(3) Both one-one and onto
(4) One-one but not onto

A function defined from real numbers to a subset $K$ of real numbers $$f(x) = \begin{cases} -x+8, & \text{if } x < 3 \\ x+2, & \text{if } x \geq 3 \end{cases}$$ Given that the function is surjective, which of the following is the set $K$?
A) $[3, \infty)$
B) $[5, \infty)$
C) $[3,5]$
D) $(-\infty, 5)$
E) $(-\infty, 3)$

The following functions are defined on the set of real numbers:
I. $f(x) = 2x - 1$ II. $g(x) = x^{2} + 2$ III. $h(x) = x^{3}$
Which of these functions are one-to-one?
A) I and II B) Only I C) I, II and III D) I and III E) Only II

Let $Z$ be the set of integers. The function $f : Z \rightarrow Z$ is defined as
$$f ( x ) = \begin{cases} x - 1 , & \text{if } x < 0 \\ x + 1 , & \text{if } x \geq 0 \end{cases}$$
Accordingly,
I. f is one-to-one. II. f is onto. III. The range of f is $Z \backslash \{ 0 \}$.
Which of these statements are true?
A) Only I
B) Only II
C) Only III
D) I and II
E) I and III