When two constants $a , b$ satisfy $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - ( a + 2 ) x + 2 a } { x ^ { 2 } - b } = 3$, what is the value of $a + b$? [2 points]
(1) $- 6$
(2) $- 4$
(3) $- 2$
(4) 0
(5) 2

Consider the function $$f ( x ) = \begin{cases} - 3 x + a & ( x \leq 1 ) \\ \frac { x + b } { \sqrt { x + 3 } - 2 } & ( x > 1 ) \end{cases}$$ If $f ( x )$ is continuous on the entire set of real numbers, find the value of $a + b$. (Here, $a$ and $b$ are constants.) [4 points]

It is known that the function $f(x)$ is an odd function defined on $\mathbb{R}$. When $x \in (-\infty, 0)$, $f(x) = 2x^3 + x^2$. Then $f(2) = $ \_\_\_\_

If $f ( x ) = \ln \left| a + \frac { 1 } { 1 - x } \right| + b$ is an odd function, then $a = $ $\_\_\_\_$ . $b = $ $\_\_\_\_$ .

Let $f : [-1,1] \rightarrow \mathbb{R}$ be a function such that $f\left(\sin\frac{x}{2}\right) = \sin x + \cos x$, for all $x \in [-\pi, \pi]$. The value of $f\left(\frac{3}{5}\right)$ is
(A) $\frac{24}{25}$
(B) $\frac{31}{25}$
(C) $\frac{33}{25}$
(D) $\frac{7}{5}$.

If $g(x) = x^{2} + x - 2$ and $\frac{1}{2}\,g\circ f(x) = 2x^{2} - 5x + 2$, then $f(x)$ is equal to
(1) $2x-3$
(2) $2x+3$
(3) $2x^{2}+3x+1$
(4) $2x^{2}-3x-1$

Let $f ( x ) = 2 ^ { 10 } x + 1$ and $g ( x ) = 3 ^ { 10 } x - 1$. If $( f \circ g ) ( x ) = x$, then $x$ is equal to:
(1) $\frac { 2 ^ { 10 } - 1 } { 2 ^ { 10 } - 3 ^ { - 10 } }$
(2) $\frac { 1 - 2 ^ { - 10 } } { 3 ^ { 10 } - 2 ^ { - 10 } }$
(3) $\frac { 3 ^ { 10 } - 1 } { 3 ^ { 10 } - 2 ^ { - 10 } }$
(4) $\frac { 1 - 3 ^ { - 10 } } { 2 ^ { 10 } - 3 ^ { - 10 } }$

If $g(x) = x ^ { 2 } + x - 1$ and $(g \circ f)(x) = 4x ^ { 2 } - 10x + 5$, then $f \left( \frac { 5 } { 4 } \right)$ is equal to
(1) $\frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $- \frac { 3 } { 2 }$

Let $f : R - \left\{ \frac { \alpha } { 6 } \right\} \rightarrow R$ be defined by $f ( x ) = \left( \frac { 5 x + 3 } { 6 x - \alpha } \right)$. Then the value of $\alpha$ for which $( f \circ f ) ( x ) = x$, for all $x \in R - \left\{ \frac { \alpha } { 6 } \right\}$, is
(1) No such $\alpha$ exists
(2) 5
(3) 8
(4) 6

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function that satisfies the relation $f ( x + y ) = f ( x ) + f ( y ) - 1 , \forall x , y \in \mathbb { R }$. If $f ^ { \prime } ( 0 ) = 2$, then $| f ( - 2 ) |$ is equal to

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) $xf'(x) - 2023f(x) = 0$

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

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

$$f ( 2 x + 5 ) = \tan \left( \frac { \pi } { 2 } x \right)$$
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) $3 \pi$

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