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.

If the domain of the function $f(x) = \frac{\sqrt{x^2 - 25}}{4 - x^2} + \log_{10}(x^2 + 2x - 15)$ is $(-\infty, \alpha) \cup (\beta, \infty)$, then $\alpha^2 + \beta^3$ is equal to:
(1) 140
(2) 175
(3) 150
(4) 125

If the domain of the function $f ( x ) = \cos ^ { - 1 } \left( \frac { 2 - | x | } { 4 } \right) + \left( \log _ { e } ( 3 - x ) \right) ^ { - 1 }$ is $[ - \alpha , \beta ) - \{ \gamma \}$, then $\alpha + \beta + \gamma$ is equal to :
(1) 12
(2) 9
(3) 11
(4) 8

If $f(x) = \frac { 4 x + 3 } { 6 x - 4 } , \quad x \neq \frac { 2 } { 3 }$ and $(f \circ f)(x) = g(x)$, where $g : \mathbb{R} - \left\{\frac { 2 } { 3 }\right\} \rightarrow \mathbb{R} - \left\{\frac { 2 } { 3 }\right\}$, then $(g \circ g \circ g)(4)$ is equal to
(1) $- \frac { 19 } { 20 }$
(2) $\frac { 19 } { 20 }$
(3) $-4$
(4) 4

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 $f: R - \{0\} \rightarrow R$ be a function satisfying $f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}$ for all $x, y$, $f(y) \neq 0$. If $f'(1) = 2024$, then
(1) $xf'(x) - 2024f(x) = 0$
(2) $xf'(x) + 2024f(x) = 0$
(3) $xf'(x) + f(x) = 2024$
(4) $xf'(x) - 2023f(x) = 0$

If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { 72 ^ { x } - 9 ^ { x } - 8 ^ { x } + 1 } { \sqrt { 2 } - \sqrt { 1 + \cos x } } , & x \neq 0 \\ a \log _ { e } 2 \log _ { e } 3 & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then the value of $a ^ { 2 }$ is equal to
(1) 968
(2) 1152
(3) 746
(4) 1250

Consider the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \frac { 2 x } { \sqrt { 1 + 9 x ^ { 2 } } }$. If the composition of $f , \underbrace { ( f \circ f \circ f \circ \cdots \circ f ) } _ { 10 \text { times } } ( x ) = \frac { 2 ^ { 10 } x } { \sqrt { 1 + 9 \alpha x ^ { 2 } } }$, then the value of $\sqrt { 3 \alpha + 1 }$ is equal to $\_\_\_\_$

Let $f : [ 0,3 ] \rightarrow \mathrm { A }$ be defined by $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 7$ and $g : [ 0 , \infty ) \rightarrow B$ be defined by $\mathrm { g } ( x ) = \frac { x ^ { 2025 } } { x ^ { 2025 } + 1 }$. If both the functions are onto and $\mathrm { S } = \{ x \in \mathbf { Z } : x \in \mathrm {~A}$ or $x \in \mathrm {~B} \}$, then $\mathrm { n } ( \mathrm { S } )$ is equal to :
(1) 29
(2) 30
(3) 31
(4) 36

Let $f ( x ) = \log _ { \mathrm { e } } x$ and $g ( x ) = \frac { x ^ { 4 } - 2 x ^ { 3 } + 3 x ^ { 2 } - 2 x + 2 } { 2 x ^ { 2 } - 2 x + 1 }$. Then the domain of $f \circ g$ is
(1) $[ 0 , \infty )$
(2) $[ 1 , \infty )$
(3) $( 0 , \infty )$
(4) $\mathbb { R }$

Let $\mathrm { A } = \{ 1,2,3,4 \}$ and $\mathrm { B } = \{ 1,4,9,16 \}$. Then the number of many-one functions $f : \mathrm { A } \rightarrow \mathrm { B }$ such that $1 \in f ( \mathrm {~A} )$ is equal to :
(1) 151
(2) 139
(3) 163
(4) 127

Let $f : \mathbb{R} - \{0\} \rightarrow \mathbb{R}$ be a function such that $f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}$. If $\lim_{x \rightarrow 0}\left(\frac{1}{\alpha x} + f(x)\right) = \beta$; $\alpha, \beta \in \mathbb{R}$, then $\alpha + 2\beta$ is equal to
(1) 5
(2) 3
(3) 4
(4) 6

Let $f : \mathbf { R } - \{ 0 \} \rightarrow ( - \infty , 1 )$ be a polynomial of degree 2, satisfying $f ( x ) f \left( \frac { 1 } { x } \right) = f ( x ) + f \left( \frac { 1 } { x } \right)$. If $f ( K ) = - 2 K$, then the sum of squares of all possible values of $K$ is :
(1) 7
(2) 6
(3) 1
(4) 9

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

$$f(x) = \frac{\left(1+x+x^{2}+x^{3}\right)(1-x)^{2}}{1-x-x^{2}+x^{3}}$$
Given this, what is the value of $f(\sqrt{2})$?
A) 1
B) 2
C) 3
D) 4
E) 5

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) $(-\infty, 3)$

$$f\left(\frac{x-1}{x+1}\right) = x^{2} - x + 2$$
Given this, what is the value of $f(3)$?
A) 5
B) 6
C) 7
D) 8
E) 11

$$\begin{aligned} & f ( x ) = x ^ { 2 } \\ & g ( x ) = 2 x - 1 \end{aligned}$$
For these functions, what is $\mathbf { g } ( \mathbf { f } ( \mathbf { 2 } ) )$?
A) 0
B) 3
C) 5
D) 7
E) 9

The following functions are given:
$f(x) = 3x - 6$
$g(x) = (x - 2)^{2}$
Accordingly, $\left(g \circ f^{-1}\right)(x)$ is equal to which of the following?
A) $\frac{3x^{2}}{2} - 1$ B) $(3x + 4)^{2}$ C) $x^{2} - 4x + 2$ D) $\frac{x^{2}}{9}$ E) $(3x - 8)^{2}$

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

$$f ( x ) = \arcsin \left( \frac { x } { 3 } + 2 \right)$$
Which of the following is the inverse function $\mathbf { f } ^ { \mathbf { - 1 } } ( \mathbf { x } )$ of this function?
A) $2 \sin ( x ) - 6$
B) $2 \sin ( x ) + 3$
C) $3 \sin ( x ) - 6$
D) $\sin ( 2 x - 6 )$
E) $\sin ( 2 x ) - 3$

The piecewise function $f : R \rightarrow R$ is defined as $f ( x ) = \left\{ \begin{array} { c l } 3 x + 1 , & x \text { is rational } \\ x ^ { 2 } , & x \text { is irrational } \end{array} \right.$
Accordingly, which of the following is $( f \circ f ) \left( \frac { \sqrt { 2 } } { 2 } \right)$?
A) $3 \sqrt { 2 } + 2$
B) $\sqrt { 2 } + 2$
C) $\frac { 1 } { 4 }$
D) $\frac { 5 } { 2 }$
E) $\frac { 7 } { 2 }$

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

$$\begin{aligned} & f ( x ) = | 2 x - 5 | \\ & g ( x ) = | x + 1 | \end{aligned}$$
The functions are given. Accordingly, what is the sum of the x values that satisfy the equation $( g \circ f ) ( x ) = 3$?
A) $-3$
B) $-1$
C) 0
D) 2
E) 5