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

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 $\_\_\_\_$

$$\begin{aligned} & f ( x ) = x ^ { 2 } \\ & g ( x ) = 2 x - 1 \end{aligned}$$
For these functions, what is $\mathbf { g } ( \mathbf { f } ( \mathbf { 2 } ) )$?
A) 0
B) 3
C) 5
D) 7
E) 9

$$f(x) = \frac{\left(1+x+x^{2}+x^{3}\right)(1-x)^{2}}{1-x-x^{2}+x^{3}}$$
Given this, what is the value of $f(\sqrt{2})$?
A) 1
B) 2
C) 3
D) 4
E) 5

The piecewise function $f : R \rightarrow R$ is defined as $f ( x ) = \left\{ \begin{array} { c l } 3 x + 1 , & x \text { is rational } \\ x ^ { 2 } , & x \text { is irrational } \end{array} \right.$
Accordingly, which of the following is $( f \circ f ) \left( \frac { \sqrt { 2 } } { 2 } \right)$?
A) $3 \sqrt { 2 } + 2$
B) $\sqrt { 2 } + 2$
C) $\frac { 1 } { 4 }$
D) $\frac { 5 } { 2 }$
E) $\frac { 7 } { 2 }$

$$\begin{aligned} & f ( x ) = | 2 x - 5 | \\ & g ( x ) = | x + 1 | \end{aligned}$$
The functions are given. Accordingly, what is the sum of the x values that satisfy the equation $( g \circ f ) ( x ) = 3$?
A) $-3$
B) $-1$
C) 0
D) 2
E) 5

$$f ( x ) = - 3 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$$
Given this, what is the product $x ^ { 3 } \cdot f \left( \frac { 1 } { x } \right)$ equal to?
A) $x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3$
B) $x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$
C) $3 x ^ { 3 } - 5 x ^ { 2 } + 2 x - 1$
D) $3 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 1$
E) $5 x ^ { 3 } - x ^ { 2 } + 2 x - 3$

The graph of the function $f : R \rightarrow R$ is given below.
Using the function f, the function g is defined for every $\mathrm { x } _ { 0 } \in \mathrm { R }$ as
$$g \left( x _ { 0 } \right) = f \left( x _ { 0 } \right) + \lim _ { x \rightarrow x _ { 0 } + } f ( x )$$
Accordingly, what is the value of (gof)(2)?
A) - 2
B) - 1
C) 0
D) 1
E) 2

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

Functions $f$ and $g$ are defined on the set of real numbers as $$\begin{aligned}& f ( x ) = \frac { x \cdot ( x - 2 ) } { 2 } \\& g ( x ) = \frac { x \cdot ( x - 1 ) \cdot ( x - 2 ) } { 3 }\end{aligned}$$ The sum of the $\mathbf{x}$ values satisfying the equality $$f ( 2 x ) = g ( x + 1 )$$ is what?\ A) 1\ B) 3\ C) 4\ D) 6\ E) 8