A function f defined on the set of real numbers satisfies the inequality
$$f ( x ) < f ( x + 2 )$$
for every real number x.
Accordingly,
I. $f ( 1 ) < f ( 5 )$ II. $| f ( - 1 ) | < | f ( 1 ) |$ III. $f ( 0 ) + f ( 2 ) < 2 \cdot f ( 4 )$
Which of these statements are always true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

For functions f and g defined on the set of real numbers
$$\begin{aligned} & f ( g ( x ) ) = x ^ { 2 } + 4 x - 1 \\ & g ( x ) = x + a \\ & f ^ { \prime } ( 0 ) = 1 \end{aligned}$$
Given this, what is a?
A) $-2$
B) $\frac { -1 } { 4 }$
C) 1
D) $\frac { 3 } { 2 }$
E) 3

$$f ( 2 x + 5 ) = \tan \left( \frac { \pi } { 2 } x \right)$$
For the function $f$ given by the equality, what is the value $f ^ { -1 } ( 1 )$?
A) $\frac { \pi } { 2 }$
B) $\frac { \pi } { 4 }$
C) $\pi$
D) $2 \pi$
E) $3 \pi$

$$f ( x ) = - 3 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$$
Given this, what is the product $x ^ { 3 } \cdot f \left( \frac { 1 } { x } \right)$ equal to?
A) $x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3$
B) $x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$
C) $3 x ^ { 3 } - 5 x ^ { 2 } + 2 x - 1$
D) $3 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 1$
E) $5 x ^ { 3 } - x ^ { 2 } + 2 x - 3$

$f : [ 1 , \infty ) \rightarrow [ 1 , \infty )$ is a function and
$$f \left( e ^ { x } \right) = \sqrt { x } + 1$$
Given this, what is the value of $f ^ { - 1 } ( 2 )$?
A) 1
B) $e - 1$
C) e
D) $e ^ { 2 }$
E) $\ln 2$

Defined on the set of real numbers R,
$$\begin{aligned} & \beta _ { 1 } = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } = 1 \right\} \\ & \beta _ { 2 } = \left\{ ( x , y ) : x ^ { 2 } + y = 2 \right\} \\ & \beta _ { 3 } = \left\{ ( x , y ) : x - y ^ { 2 } = 3 \right\} \end{aligned}$$
Which of these relations define a function of the form $\mathbf { y } = \mathbf { f } ( \mathbf { x } )$ on $R$?
A) Only $\beta _ { 1 }$
B) Only $\beta _ { 2 }$
C) $\beta _ { 1 }$ and $\beta _ { 3 }$
D) $\beta _ { 2 }$ and $\beta _ { 3 }$
E) $\beta _ { 1 } , \beta _ { 2 }$ and $\beta _ { 3 }$

I. $f ( x ) = 2 x$ II. $f ( x ) = 2 ^ { x }$ III. $f ( x ) = x ^ { 2 }$ Which of these functions satisfy the equation $f ( a + b ) = f ( a ) \cdot f ( b )$ for every real number a and b?
A) Only I
B) Only II
C) I and II
D) I and III
E) II and III

$$\begin{aligned} & f ( x ) = - \log _ { 2 } x \\ & g ( x ) = \log _ { 10 } x \end{aligned}$$
Given this, what is the value of a that satisfies the equality $\left( \right.$ gof $\left. ^ { - 1 } \right) ( a ) = \ln 2$?
A) $\ln 2$
B) $\frac { \ln 2 } { \ln 10 }$
C) $\frac { \ln 10 } { \ln 2 }$
D) $\ln \left( \frac { 1 } { 10 } \right)$
E) $\ln \left( \frac { 1 } { 2 } \right)$

Let $A = \{ 1,2,3 \}$ and $f : A \rightarrow A$ be a function. How many one-to-one functions $f$ satisfy the condition
$$f ( n ) \neq n$$
for every $n \in A$?
A) 1
B) 2
C) 3
D) 4
E) 5

Below are the graph of the line $y = x$ and the graph of the function $y = f ( x )$.
Starting from point $\mathbf { Q } ( \mathbf { a } , \mathbf { 0 } )$ and following the arrows, point $\mathbf { P } ( \mathbf { a } , \mathbf { b } )$ is reached. Accordingly, $\mathbf { b }$ is equal to which of the following?
A) $a + f ( a )$
B) $a \cdot f ( a )$
C) $f ( a ) - a$
D) $f ( f ( a ) )$
E) $f ( a + f ( a ) )$

For the function $f ( x ) = \log _ { x } 2$,
$$f \left( 4 ^ { a } \right) \cdot f ^ { - 1 } \left( \frac { 1 } { 3 } \right) = 6$$
What is the value of a that satisfies this equation?
A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 4 } { 3 }$

The graph of the function $f : R \rightarrow R$ is given below.
Using the function f, the function g is defined for every $\mathrm { x } _ { 0 } \in \mathrm { R }$ as
$$g \left( x _ { 0 } \right) = f \left( x _ { 0 } \right) + \lim _ { x \rightarrow x _ { 0 } + } f ( x )$$
Accordingly, what is the value of (gof)(2)?
A) - 2
B) - 1
C) 0
D) 1
E) 2

For functions $f$ and $g$ defined on the set of positive real numbers
$$\begin{aligned} & ( f \circ g ) ( x ) = f ( x ) \cdot g ( x ) \\ & f ( x ) = 2 x + 3 \end{aligned}$$
Given that, what is the value of $\mathbf { g } ( \mathbf { 1 } )$?
A) 1 B) 2 C) 3 D) 4 E) 5

Let $A , B$ be two sets, $B \backslash A \neq \emptyset$ and the Cartesian product set $( A \backslash B ) \times A$ has 14 elements.
Accordingly, what is the minimum number of elements in set B?
A) 1
B) 3
C) 4
D) 6
E) 8

Let $\mathrm { P } ( \mathrm { x } )$ be a second-degree polynomial and $\mathrm { Q } ( \mathrm { x } ) = \mathrm { k }$ be a constant polynomial such that
$$\begin{aligned} & P ( x ) + Q ( x ) = 2 x ^ { 2 } + 3 \\ & P ( Q ( x ) ) = 9 \end{aligned}$$
Accordingly, what is the sum of the values that k can take?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 3 } { 4 }$

For a function f defined on the set of positive real numbers with $f ( 3 ) = 2$, the derivative of the function f is given as
$$f ^ { \prime } ( x ) = x ^ { 2 } + x$$
For the function $\mathbf { g } ( \mathbf { x } ) = \mathbf { f } ^ { - \mathbf { 1 } } ( \mathbf { 2 x } )$, what is the value of $\mathbf { g } ^ { \prime } ( \mathbf { 1 } )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 6 }$

Functions $f$ and $g$ defined on the set of real numbers satisfy the equalities
$$\begin{aligned} & ( f + g ) ( x ) = x ^ { 2 } \\ & ( f - g ) ( 2 x ) = x \end{aligned}$$
Accordingly, what is the product $f ( 4 ) \cdot g ( 4 )$?
A) 45
B) 51
C) 54
D) 60
E) 63

Functions $f$ and $g$ with domain of integers are defined as
$$\begin{aligned} & f ( n ) = n + \frac { 1 } { 3 } \\ & g ( n ) = n + \frac { 1 } { 6 } \end{aligned}$$
Given this, I. $f \circ f \circ f$ II. $f \circ g \circ f$ III. $g \circ f \circ g$ For which of these functions does the image set consist only of integers?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

Function f is defined for every $\mathrm { x } \in ( 0,3 ]$ as
$$f ( x ) = 2 x + 1$$
and satisfies the equality
$$f ( x ) = f ( x + 3 )$$
for every real number x. Accordingly, what is the sum $\mathbf { f } ( \mathbf { 6 } ) + \mathbf { f } ( \mathbf { 7 } ) + \mathbf { f } ( \mathbf { 8 } )$?
A) 8
B) 12
C) 15
D) 18
E) 21

Let k be a real number. The functions f and g defined on the set of positive real numbers are
$$\begin{aligned} & f ( x ) = k x ^ { 2 } + 1 \\ & g ( x ) = \sqrt { x } + 2 \end{aligned}$$
defined in the form.
$$( f \circ g ) ( 9 ) = 6$$
Given that, what is the value of f(2)?
A) $\frac { 7 } { 5 }$ B) $\frac { 8 } { 5 }$ C) $\frac { 9 } { 5 }$ D) 2 E) 3

Functions $f$ and $g$ are defined on the set of real numbers as $$\begin{aligned}& f ( x ) = \frac { x \cdot ( x - 2 ) } { 2 } \\& g ( x ) = \frac { x \cdot ( x - 1 ) \cdot ( x - 2 ) } { 3 }\end{aligned}$$ The sum of the $\mathbf{x}$ values satisfying the equality $$f ( 2 x ) = g ( x + 1 )$$ is what?\ A) 1\ B) 3\ C) 4\ D) 6\ E) 8

A function $f$ on the set of real numbers is defined as $$f ( x ) = \frac { | x | } { 1 + | x | }$$ Accordingly, which of the following is the image set of the interval $[ - 2,1 )$ under the function $\mathbf{f}$?\ A) $[ 0,1 ]$\ B) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right]$\ C) $\left[ \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$\ D) $\left[ 0 , \frac { 1 } { 3 } \right]$\ E) $\left[ 0 , \frac { 2 } { 3 } \right]$

Let $a$ and $b$ be non-zero real numbers. A function $f$ defined on the set of real numbers
$$\begin{aligned} & f ( a x + b ) = x \\ & f ( a ) = \frac { b } { a } \end{aligned}$$
satisfies the equalities.
Accordingly, what is the value of $\mathrm { f } ( 0 )$?
A) $\frac { - 1 } { 2 }$ B) $\frac { - 1 } { 3 }$ C) $\frac { - 2 } { 3 }$ D) 1 E) 2

A function f defined on the set of real numbers satisfies the equality
$$f ( x + y ) = f ( x ) + f ( y )$$
for every real numbers x and y. Given that $\mathbf { f } ( \mathbf { 2 } ) - \mathbf { f } ( \mathbf { 1 } ) = \mathbf { 1 0 }$,
what is the result of the operation $$\frac { f ( 3 ) \cdot f ( 4 ) } { f ( 5 ) }$$?
A) 15
B) 16
C) 18
D) 21
E) 24

Let $a$ and $b$ be real numbers. The functions f and g are defined on the set of real numbers as
$$\begin{aligned} & f(x) = ax - b \\ & g(x) = bx - 2 \end{aligned}$$
Given that
$$\begin{aligned} & (f + g)(1) = f(1) \\ & (f + g)(2) = g(2) \end{aligned}$$
what is the product $\mathbf{a} \cdot \mathbf{b}$?
A) 2
B) 4
C) 6
D) 8
E) 10