12. If $f ( x ) = \left\{ \begin{aligned} \ln x & \text { for } 0 < x \leq 2 \\ x ^ { 2 } \ln 2 & \text { for } 2 < x \leq 4 , \end{aligned} \right.$ then $\lim _ { x \rightarrow 2 } f ( x )$ is
(A) $\ln 2$
(B) $\ln 8$
(C) $\ln 16$
(D) 4
(E) nonexistent [Figure]

For a function $y = f ( x )$ defined on the closed interval $[ 0,5 ]$, define the function $g ( x )$ as $$g ( x ) = \begin{cases} \{ f ( x ) \} ^ { 2 } & ( 0 \leqq x \leqq 3 ) \\ ( f \circ f ) ( x ) & ( 3 < x \leqq 5 ) \end{cases}$$ Which of the following graphs of the function $y = f ( x )$ make the function $g ( x )$ continuous on the closed interval $[ 0,5 ]$? Select all that apply from . [4 points] ㄱ. [graph] ㄴ. [graph] ㄷ. [graph]
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

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

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

102- Two functions with the sets $g=\{(2,5),(3,4),(1,6),(4,7),(8,1)\}$ and $f(x)=2x-5$ are given. If $(f^{-1}\circ g)(a)=6$, what is $a$?
(1) $1$ (2) $2$ (3) $3$ (4) $4$

112- For which value of $a$ is the function $$f(x) = \begin{cases} \dfrac{a(1+\sqrt[5]{1-x})}{x^2 - 2x} & ; \ x > 2 \\[8pt] x - a & ; \ x \leq 2 \end{cases}$$ always continuous?

p{6cm}} (2) $1.6$	(1) $1.2$
[6pt] (4) $2.2$	(3) $2.4$

114. Suppose $f(x) = \begin{cases} -1 & x < -1 \\ x & -1 \leq x \leq 1 \\ 1 & x > 1 \end{cases}$ and $g(x) = 1 - x^2$. The number of elements of the set of points where $g \circ f$ and $f \circ g$ are not differentiable is:
(1) $2$ (2) $3$ (3) $4$ (4) $5$

110. If $f(x) = \dfrac{\sqrt{3}x}{3x - \sqrt{x}}$, what is $fofof(\sqrt{2})$?
(1) $\dfrac{1}{\sqrt{2}}$ (2) $\sqrt{2}$ (3) $2$ (4) $\dfrac{1}{2}$

For every real number $x \neq - 1$, let $f ( x ) = \frac { x } { x + 1 }$. Write $f _ { 1 } ( x ) = f ( x )$ and for $n \geq 2 , f _ { n } ( x ) = f \left( f _ { n - 1 } ( x ) \right)$. Then,
$$f _ { 1 } ( - 2 ) \cdot f _ { 2 } ( - 2 ) \cdots \cdots f _ { n } ( - 2 )$$
must equal
(A) $\frac { 2 ^ { n } } { 1 \cdot 3 \cdot 5 \cdots \cdot ( 2 n - 1 ) }$
(B) 1
(C) $\frac { 1 } { 2 } \binom { 2 n } { n }$
(D) $\binom { 2 n } { n }$.

Let $f ( x ) = x ^ { 2 }$ and $g ( x ) = \sin x$ for all $x \in \mathbb { R }$. Then the set of all $x$ satisfying $( f \circ g \circ g \circ f ) ( x ) = ( g \circ g \circ f ) ( x )$, where $( f \circ g ) ( x ) = f ( g ( x ) )$, is
(A) $\pm \sqrt { n \pi } , n \in \{ 0,1,2 , \ldots \}$
(B) $\pm \sqrt { n \pi } , n \in \{ 1,2 , \ldots \}$
(C) $\frac { \pi } { 2 } + 2 n \pi , n \in \{ \ldots , - 2 , - 1,0,1,2 , \ldots \}$
(D) $2 n \pi , n \in \{ \ldots , - 2 , - 1,0,1,2 , \ldots \}$

If $f ( x )$ is continuous and $f \left( \frac { 9 } { 2 } \right) = \frac { 2 } { 9 }$, then $\lim _ { x \rightarrow 0 } f \left( \frac { 1 - \cos 3 x } { x ^ { 2 } } \right)$ equals to
(1) $\frac { 8 } { 9 }$
(2) 0
(3) $\frac { 2 } { 9 }$
(4) $\frac { 9 } { 2 }$

Let $f(x) = x^2$, $g(x) = \sin x$ for all $x \in \mathbb{R}$ and $h(x) = (gof)(x) = g(f(x))$. Statement I: $h$ is not differentiable at $x = 0$. Statement II: $(hog)(x) = \sin^2(\sin x)$. Which of the following is correct?
(1) Statement I is false, Statement II is true
(2) Statement I is true, Statement II is false
(3) Both Statement I and Statement II are true
(4) Both Statement I and Statement II are false

For $x \in \mathbb { R }$, $f ( x ) = | \log 2 - \sin x |$ and $g ( x ) = f ( f ( x ) )$, then:
(1) $g$ is not differentiable at $x = 0$
(2) $g ^ { \prime } ( 0 ) = \cos ( \log 2 )$
(3) $g ^ { \prime } ( 0 ) = - \cos ( \log 2 )$
(4) $g$ is differentiable at $x = 0$ and $g ^ { \prime } ( 0 ) = - \sin ( \log 2 )$

If $f(x) = \log_e\frac{1-x}{1+x}$, $|x| < 1$, then $f\left(\frac{2x}{1+x^2}\right)$ is equal to
(1) $f(x^2)$
(2) $2f(x^2)$
(3) $-2f(x)$
(4) $2f(x)$

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x) = \log _ { \mathrm { e } } ( x ^ { 2 } + 1 ) - e ^ { - x } + 1$ and $g(x) = \frac { 1 - 2 e ^ { 2 x } } { e ^ { x } }$. Then, for which of the following range of $\alpha$, the inequality $f\left( g\left( \frac { ( \alpha - 1 ) ^ { 2 } } { 3 } \right) \right) > f\left( g\left( \alpha - \frac { 5 } { 3 } \right) \right)$ holds?
(1) $( - 2 , - 1 )$
(2) $(2, 3)$
(3) $(1, 2)$
(4) $( - 1, 1 )$

Let $f ( x ) = \frac { x - 1 } { x + 1 } , x \in R - \{ 0 , - 1 , 1 \}$. If $f ^ { n + 1 } ( x ) = f \left( f ^ { n } ( x ) \right)$ for all $n \in N$, then $f ^ { 6 } ( 6 ) + f ^ { 7 } ( 7 )$ is equal to
(1) $\frac { 7 } { 6 }$
(2) $- \frac { 3 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $- \frac { 11 } { 12 }$

If $f ( x ) = \left\{ \begin{array} { l l } x + a , & x \leq 0 \\ | x - 4 | , & x > 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x + 1 , & x < 0 \\ ( x - 4 ) ^ { 2 } + b , & x \geq 0 \end{array} \right.$ are continuous on $R$, then $( g \circ f ) ( 2 ) + ( f \circ g ) ( - 2 )$ is equal to:
(1) $- 10$
(2) 10
(3) 8
(4) $- 8$

For some $a , b , c \in \mathbb { N }$, let $f ( x ) = a x - 3$ and $g ( x ) = x ^ { b } + c , x \in \mathbb { R }$. If $( f \circ g ) ^ { - 1 } ( x ) = \left( \frac { x - 7 } { 2 } \right) ^ { \frac { 1 } { 3 } }$, then $( f \circ g ) ( a c ) + ( g \circ f ) ( b )$ is equal to $\_\_\_\_$ .

Let $f$, $g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as $f(x) = \left\{ \begin{array}{cc} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{array} \right.$, $\quad g(x) = \left\{ \begin{array}{cc} \frac{\sin(x+1)}{(x+1)}, & x \neq -1 \\ 1, & x = -1 \end{array} \right.$ and $h(x) = 2[x] - f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim_{x \rightarrow 1} g(h(x-1))$ is
(1) 1
(2) $\sin(1)$
(3) $-1$
(4) 0

Let $f(x) = \begin{cases} x-1, & x \text{ is even,} \\ 2x, & x \text{ is odd,} \end{cases}$ $x \in \mathbb{N}$. If for some $a \in \mathbb{N}$, $f(f(f(a))) = 21$, then $\lim_{x \to a^-} \left(\frac{x^3}{a} - \left\lfloor\frac{x}{a}\right\rfloor\right)$, where $\lfloor t \rfloor$ denotes the greatest integer less than or equal to $t$, is equal to:
(1) 121
(2) 144
(3) 169
(4) 225

If $f(x) = \frac { 4 x + 3 } { 6 x - 4 } , \quad x \neq \frac { 2 } { 3 }$ and $(f \circ f)(x) = g(x)$, where $g : \mathbb{R} - \left\{\frac { 2 } { 3 }\right\} \rightarrow \mathbb{R} - \left\{\frac { 2 } { 3 }\right\}$, then $(g \circ g \circ g)(4)$ is equal to
(1) $- \frac { 19 } { 20 }$
(2) $\frac { 19 } { 20 }$
(3) $-4$
(4) 4

Consider the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \frac { 2 x } { \sqrt { 1 + 9 x ^ { 2 } } }$. If the composition of $f , \underbrace { ( f \circ f \circ f \circ \cdots \circ f ) } _ { 10 \text { times } } ( x ) = \frac { 2 ^ { 10 } x } { \sqrt { 1 + 9 \alpha x ^ { 2 } } }$, then the value of $\sqrt { 3 \alpha + 1 }$ is equal to $\_\_\_\_$

Q85. Consider the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \frac { 2 x } { \sqrt { 1 + 9 x ^ { 2 } } }$. If the composition of $f , \underbrace { ( f \circ f \circ f \circ \cdots \circ f ) } _ { 10 \text { times } } ( x ) = \frac { 2 ^ { 10 } x } { \sqrt { 1 + 9 \alpha x ^ { 2 } } }$, then the value of $\sqrt { 3 \alpha + 1 }$ is equal to $\_\_\_\_$

Q67. Let $f ( x ) = x ^ { 2 } + 9 , g ( x ) = \frac { x } { x - 9 }$ and $\mathrm { a } = f \circ g ( 10 ) , \mathrm { b } = g \circ f ( 3 )$. If e and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac { x ^ { 2 } } { a } + \frac { y ^ { 2 } } { b } = 1$, then $8 \mathrm { e } ^ { 2 } + l ^ { 2 }$ is equal to.
(1) 8
(2) 16
(3) 6
(4) 12

2. For ALL APPLICANTS.

Let
$$f ( x ) = x + 1 \quad \text { and } \quad g ( x ) = 2 x$$
We will, for example, write $f g$ to denote the function "perform $g$ then perform $f$ " so that
$$f g ( x ) = f ( g ( x ) ) = 2 x + 1$$
If $i \geqslant 0$ is an integer we will, for example, write $f ^ { i }$ to denote the function which performs $f i$ times, so that
$$f ^ { i } ( x ) = \underbrace { f f f \cdots f } _ { i \text { times } } ( x ) = x + i .$$
(i) Show that
$$f ^ { 2 } g ( x ) = g f ( x )$$
(ii) Note that
$$g f ^ { 2 } g ( x ) = 4 x + 4$$
Find all the other ways of combining $f$ and $g$ that result in the function $4 x + 4$.
(iii) Let $i , j , k \geqslant 0$ be integers. Determine the function
$$f ^ { i } g f ^ { j } g f ^ { k } ( x )$$
(iv) Let $m \geqslant 0$ be an integer. How many different ways of combining the functions $f$ and $g$ are there that result in the function $4 x + 4 m$ ?