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 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

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

$$f ( x ) = \left\{ \begin{array} { c c } \frac { a x } { x + 2 b } \cdot \cot x & , x \neq 0 \\ 2 & , x = 0 \end{array} \right.$$
The function is continuous at the point $x = 0$. Accordingly, what is the ratio $\frac { a } { b }$?
A) 1
B) 2
C) 4
D) $\frac { 1 } { 3 }$
E) $\frac { 1 } { 6 }$

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

Sets $A$, $B$, and $C$ are defined as $$\begin{aligned}& A = \{ ( x , x ) : x \in \mathbb { R } \} \\& B = \{ ( x , 3 - x ) : x \in \mathbb { R } \} \\& C = \{ ( x , x + 4 ) : x \in \mathbb { R } \}\end{aligned}$$ Given that $( p , q ) \in A \cap B$ and $( r , s ) \in B \cap C$, $$\frac { p - r } { q + s }$$ what is the value of this expression?\ A) $\frac { 1 } { 3 }$\ B) $\frac { 1 } { 4 }$\ C) $\frac { 3 } { 4 }$\ D) $\frac { 4 } { 5 }$\ E) $\frac { 2 } { 5 }$

A function $f$ on the set of real numbers is defined for every real number $x$ where $n$ is an integer as $$f ( x ) = x - n , \quad x \in [ n , n + 1 )$$ Accordingly, $$f ( 1 ) + f \left( \frac { 7 } { 3 } \right) + f \left( \frac { 13 } { 6 } \right)$$ what is this sum?\ A) $\frac { 1 } { 2 }$\ B) $\frac { 2 } { 3 }$\ C) $\frac { 7 } { 6 }$\ D) 1\ E) 2

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

$$f ( x ) = \left\{ \begin{array} { l l l } 10 - x ^ { 2 } & , & x < 0 \\ a x + b & , & 0 \leq x \leq 3 \\ ( 1 - x ) ^ { 2 } & , & x > 3 \end{array} \right.$$
The function is continuous on the set of real numbers.
Accordingly, what is the sum $\mathbf { a } + \mathbf { b }$?
A) 16 B) 15 C) 12 D) 9 E) 8

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

A function f is defined on a subset of the set of real numbers as
$$f ( x ) = \frac { x ^ { 2 } - 4 x + 4 } { x - 2 } + \frac { x ^ { 2 } - 6 x + 9 } { 2 x - 6 }$$
Accordingly, $$\lim _ { x \rightarrow 2 } f ( x ) + \lim _ { x \rightarrow 3 } f ( x )$$
what is the value of this expression?
A) $\frac { 3 } { 2 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 4 }$

Let a be a real number. A function f is defined on the set of real numbers as
$$f ( x ) = \left\{ \begin{array} { c c c } a - x & , & x < 1 \\ 5 x - 4 & , & 1 \leq x \leq 5 \\ ( x - a ) ^ { 2 } + 12 & , & x > 5 \end{array} \right.$$
If there is only one point where the function f is not continuous, what is the value of
$$f ( 7 ) - f ( 0 )$$?
A) 3
B) 4
C) 5
D) 6
E) 7

Let $a$ and $b$ be non-zero integers. A function $f$ is defined on the set of real numbers as
$$f ( x ) = a x + b$$
$$( f \circ f ) ( x ) = f ( x + 2 ) + f ( x )$$
According to this, what is the value of $f(3)$?
A) 7
B) 8
C) 9
D) 10
E) 11

For functions $f$ and $g$ defined on the set of real numbers $$\begin{aligned} & (f \circ g)(x) = x^2 + 3x + 1 \\ & (g \circ f)(x) = x^2 - x + 1 \end{aligned}$$ the equalities are satisfied. Given that $f(2) = 1$, what is the value of $f(3)$?
A) 5
B) 6
C) 7
D) 8
E) 9

In the rectangular coordinate plane, the graph of the function $f(x)$ defined on the closed interval $[0,5]$ is given in the figure.
If the function $(f \circ f \circ f)(x)$ attains its maximum value at the point $x = a$, in which of the following open intervals is the number $a$?
A) $( 0,1 )$
B) $( 1,2 )$
C) $( 2,3 )$
D) $( 3,4 )$
E) $( 4,5 )$

Let $a$ and $b$ be real numbers. A function $f$ that is continuous on the set of real numbers is defined as
$$f ( x ) = \begin{cases} x ^ { 2 } - 4 & , x \leq a \\ 5 x - 8 & , a < x \leq b \\ 7 & , x > b \end{cases}$$
Accordingly, what is the sum $a + b$?
A) 4
B) 5
C) 6
D) 7
E) 8

Let $a$ and $b$ be real numbers. For functions $f$ and $g$ defined on the set of real numbers
$$\begin{aligned} & f(x) = x^{2} + ax + b \\ & g(x) = ax + 2 \\ & (f + g)(3) = 4 \\ & (f - g)(5) = 6 \end{aligned}$$
These equalities are satisfied.
Accordingly, what is the difference $\mathrm{a} - \mathrm{b}$?
A) 17 B) $\frac{52}{3}$ C) 18 D) $\frac{56}{3}$ E) 19

Let $a$ be a positive real number. The functions f and g are defined on the set of real numbers as
$$\begin{aligned} & f(x) = x + a \\ & g(x) = ax + 1 \end{aligned}$$
Given that $(\mathbf{f} \cdot \mathbf{g})(\mathbf{1}) = (\mathbf{f} + \mathbf{g})(\mathbf{2})$, what is $\mathbf{g}(\mathbf{7})$?
A) 8 B) 15 C) 22 D) 29 E) 36

Functions $f$ and $g$ are defined on the set of real numbers as
$$\begin{aligned} & f(x) = \frac{3x + 4}{2} \\ & g(x) = \frac{2x - 4}{3} \end{aligned}$$
If $(\mathbf{f} \circ \mathbf{g})(\mathbf{a}) = \mathbf{f}(\mathbf{a}) = \mathbf{b}$, what is the product $\mathbf{a} \cdot \mathbf{b}$?
A) $-20$ B) $-12$ C) $-8$ D) 4 E) 16

Let $a$ and $b$ be real numbers. For the functions $f$ and $g$ defined on the set of real numbers as
$$\begin{aligned} & f(x) = \frac{x}{2} + 1 \\ & g(x) = 2x - 3 \end{aligned}$$
the equalities
$$\begin{aligned} & (f + g)(a) = f(a) \\ & (f - g)(b) = g(b) \end{aligned}$$
are satisfied. Accordingly, what is the value of $(f \circ g)(a \cdot b)$?
A) $\frac{1}{2}$ B) $\frac{5}{2}$ C) $\frac{9}{2}$ D) $\frac{13}{2}$ E) $\frac{17}{2}$

Let $a$ be a non-zero real number, and $b$ and $c$ be real numbers. For the function $f(x) = ax + b$ defined on the set of real numbers and its inverse function $f^{-1}$,
$$\begin{aligned} & \lim_{x \rightarrow b} \frac{f(x)}{f^{-1}(x)} = c \\ & f(1) = 3 \end{aligned}$$
are given. Accordingly, what is the sum of the different values that $c$ can take?
A) 6 B) 7 C) 10 D) 11 E) 14