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 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 $a$ 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 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