Let $f$ be the function defined above. $$f ( x ) = \begin{cases} \frac { ( 2 x + 1 ) ( x - 2 ) } { x - 2 } & \text { for } x \neq 2 \\ k & \text { for } x = 2 \end{cases}$$ For what value of $k$ is $f$ continuous at $x = 2$ ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5

Let $f ( x ) = ( 2 x + 1 ) ^ { 3 }$ and let $g$ be the inverse function of $f$. Given that $f ( 0 ) = 1$, what is the value of $g ^ { \prime } ( 1 )$ ?
(A) $- \frac { 2 } { 27 }$
(B) $\frac { 1 } { 54 }$
(C) $\frac { 1 } { 27 }$
(D) $\frac { 1 } { 6 }$
(E) 6

Let $f$ be the function defined by $f ( x ) = \sqrt { | x - 2 | }$ for all $x$. Which of the following statements is true?
(A) $f$ is continuous but not differentiable at $x = 2$.
(B) $f$ is differentiable at $x = 2$.
(C) $f$ is not continuous at $x = 2$.
(D) $\lim _ { x \rightarrow 2 } f ( x ) \neq 0$
(E) $x = 2$ is a vertical asymptote of the graph of $f$.

If the function $f$ is continuous at $x = 3$, which of the following must be true?
(A) $f ( 3 ) < \lim _ { x \rightarrow 3 } f ( x )$
(B) $\lim _ { x \rightarrow 3 ^ { - } } f ( x ) \neq \lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(C) $f ( 3 ) = \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = \lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(D) The derivative of $f$ at $x = 3$ exists.
(E) The derivative of $f$ is positive for $x < 3$ and negative for $x > 3$.

Many physiological and biochemical processes, such as heartbeats and respiration rate, present scales constructed from the relationship between surface area and mass (or volume) of the animal. One of these scales, for example, considers that ``the cube of the surface area $S$ of a mammal is proportional to the square of its mass $M$''.
This is equivalent to saying that, for a constant $k > 0$, the area $S$ can be written as a function of $M$ by means of the expression:
(A) $S = k \cdot M$ (B) $S = k \cdot M^{\frac{1}{3}}$ (C) $S = k^{\frac{1}{3}} \cdot M^{\frac{1}{3}}$ (D) $S = k^{\frac{1}{3}} \cdot M^{\frac{2}{3}}$ (E) $S = k^{\frac{1}{3}} \cdot M^{2}$

A function $f$ is defined by $f(x) = 2x + 5$. What is the value of $f^{-1}(11)$?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Cortisol is a hormone produced by the adrenal glands and can be considered an important marker of physiological stress. In a study conducted with nurses, it was found that the concentration of salivary cortisol on a work day, denoted by $T$, was, on average, 1.59 times the concentration of salivary cortisol on a day off, denoted by $F$.
In this study, the relationship obtained between $T$ and $F$ was
(A) $T = 1.59 + F$
(B) $F = 1.59 + T$
(C) $\dfrac{T}{F} = 1.59$
(D) $\dfrac{F}{T} = 1.59$
(E) $F \cdot T = 1.59$

For sets $A$ and $B$, let $f : A \rightarrow B$ and $g : B \rightarrow A$ be functions such that $f ( g ( x ) ) = x$ for each $x$. For each statement below, write whether it is TRUE or FALSE. a) The function $f$ must be one-to-one.
Answer: $\_\_\_\_$ b) The function $f$ must be onto.
Answer: $\_\_\_\_$ c) The function $g$ must be one-to-one.
Answer: $\_\_\_\_$ d) The function $g$ must be onto.
Answer: $\_\_\_\_$

Given a real number $x$, define $g ( x ) = x ^ { 2 } e ^ { x }$ if $x \geq 0$ and $g ( x ) = x e ^ { - x }$ if $x < 0$.
(A) The function $g$ is continuous everywhere.
(B) The function $g$ is differentiable everywhere.
(C) The function $g$ is one-to-one.
(D) The range of $g$ is the set of all real numbers.

[14 points] Let $\mathbb { R } _ { + }$ denote the set of positive real numbers. A one-to-one and onto function $f : \mathbb { R } _ { + } \rightarrow \mathbb { R } _ { + }$ is called golden if $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R } _ { + }$.
(i) Find all golden functions (if any) of the form $f ( x ) = a x ^ { b }$. Find all golden functions (if any) of the form $f ( x ) = a b ^ { x }$. In both cases $a$ and $b$ are suitable real numbers.
(ii) Show that there is no one-to-one and onto function $f : \mathbb { R } \rightarrow \mathbb { R }$ such that $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R }$.

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

For a function $y = f ( x )$ defined on all real numbers, let $N ( f )$ denote the smallest natural number $k$ such that the function $y = x ^ { k } f ( x )$ is continuous at $x = 0$. For example,
$$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { x } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{array} \text { then } N ( f ) = 2 \right. \text { . }$$
For the following functions $g _ { i } ( i = 1,2,3 )$, let $N \left( g _ { i } \right) = a _ { i }$. Which correctly represents the order of $a _ { i }$? [3 points]
$$\begin{aligned} & g _ { 1 } ( x ) = \begin{cases} \frac { | x | } { x } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \\ & g _ { 2 } ( x ) = \begin{cases} - x ^ { 2 } + 1 & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \\ & g _ { 3 } ( x ) = \begin{cases} \frac { 1 } { x ^ { 2 } } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \end{aligned}$$
(1) $a _ { 1 } = a _ { 2 } < a _ { 3 }$
(2) $a _ { 1 } < a _ { 2 } = a _ { 3 }$
(3) $a _ { 1 } = a _ { 2 } = a _ { 3 }$
(4) $a _ { 2 } = a _ { 3 } < a _ { 1 }$
(5) $a _ { 3 } < a _ { 1 } = a _ { 2 }$

When the graph of the function $y = f ( x )$ is as shown in the figure, which of the following statements are correct? [4 points]
ㄱ. $\lim _ { x \rightarrow +0 } f ( x ) = 1$ ㄴ. $\lim _ { x \rightarrow 1 } f ( x ) = f ( 1 )$ ㄷ. The function $( x - 1 ) f ( x )$ is continuous at $x = 1$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Find the value of $\lim_{x \rightarrow 2} \frac{(x-2)(x+3)}{x-2}$. [3 points]

For a quadratic function $f ( x )$ with leading coefficient 1 and the function $$g ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { \ln ( x + 1 ) } & ( x \neq 0 ) \\ 8 & ( x = 0 ) \end{array} \right.$$ when the function $f ( x ) g ( x )$ is continuous on the interval $( - 1 , \infty )$, what is the value of $f ( 3 )$? [3 points]
(1) 6
(2) 9
(3) 12
(4) 15
(5) 18

The figure shows a function $f : X \rightarrow X$. What is the value of $f ( 2 ) + f ^ { - 1 } ( 2 )$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7

On the coordinate plane, when the graph of the function $y = \frac { 3 } { x - 5 } + k$ is symmetric with respect to the line $y = x$, what is the value of the constant $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a real number $k$, let $g ( x )$ be the inverse function of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x + k$. For the equation $4 f ^ { \prime } ( x ) + 12 x - 18 = \left( f ^ { \prime } \circ g \right) ( x )$ to have a real root in the closed interval $[ 0,1 ]$, let $m$ be the minimum value of $k$ and $M$ be the maximum value of $k$. Find the value of $m ^ { 2 } + M ^ { 2 }$. [4 points]

The figure shows two functions $f : X \rightarrow Y , g : Y \rightarrow Z$. Find the value of $( g \circ f ) ( 2 )$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

There are two functions $f ( x ) , g ( x )$ differentiable on the set of all real numbers. $f ( x )$ is the inverse function of $g ( x )$ and $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$. If the function $h ( x ) = x g ( x )$, what is the value of $h ^ { \prime } ( 2 )$? [3 points]
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 5 } { 3 }$
(4) 2
(5) $\frac { 7 } { 3 }$

As shown in the figure, the graph of the function $f ( x )$ defined on the closed interval $[ 0,4 ]$ is formed by connecting the points $( 0,0 ) , ( 1,4 ) , ( 2,1 ) , ( 3,4 ) , ( 4,3 )$ in order with line segments. Find the number of sets $X = \{ a , b \}$ satisfying the following condition. (Here, $0 \leq a < b \leq 4$) [4 points]
A function $g ( x ) = f ( f ( x ) )$ from $X$ to $X$ exists and satisfies $g ( a ) = f ( a ) , g ( b ) = f ( b )$.
(1) 11
(2) 13
(3) 15
(4) 17
(5) 19

The figure shows a function $f : X \rightarrow X$. [Figure] What is the value of $f ( 4 ) + ( f \circ f ) ( 2 )$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7

When the inverse function of $f ( x ) = \frac { 1 } { 1 + e ^ { - x } }$ is $g ( x )$, what is the value of $g ^ { \prime } ( f ( - 1 ) )$? [3 points]
(1) $\frac { 1 } { ( 1 + e ) ^ { 2 } }$
(2) $\frac { e } { 1 + e }$
(3) $\left( \frac { 1 + e } { e } \right) ^ { 2 }$
(4) $\frac { e ^ { 2 } } { 1 + e }$
(5) $\frac { ( 1 + e ) ^ { 2 } } { e }$

The figure shows two functions $f : X \rightarrow X , g : X \rightarrow X$. What is the value of $( g \circ f ) ( 1 )$? [3 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

For the function $f ( x ) = \frac { k } { x - 3 } + 1$, when $f ^ { - 1 } ( 7 ) = 4$, what is the value of the constant $k$? (Note: $k \neq 0$) [3 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10