For a polynomial function $f ( x )$, define the function $g ( x )$ as follows: $$g ( x ) = \begin{cases} x & ( x < - 1 \text{ or } x > 1 ) \\ f ( x ) & ( - 1 \leq x \leq 1 ) \end{cases}$$ For the function $h ( x ) = \lim _ { t \rightarrow 0 + } g ( x + t ) \times \lim _ { t \rightarrow 2 + } g ( x + t )$, which of the following statements in the given options are correct? [4 points]
ㄱ. $h ( 1 ) = 3$ ㄴ. The function $h ( x )$ is continuous on the set of all real numbers. ㄷ. If the function $g ( x )$ is decreasing on the closed interval $[ - 1, 1 ]$ and $g ( - 1 ) = - 2$, then the function $h ( x )$ has a minimum value on the set of all real numbers.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ

2. Which of the following functions is an odd function?
A. $y = \sqrt { x }$
B. $y = | \sin x |$
C. $y = \cos x$
D. $y = e ^ { x } - e ^ { - x }$

Given that the domain of function $f ( x )$ is $\mathbb { R }$, $f ( x + 2 )$ is an even function, and $f ( 2 x + 1 )$ is an odd function, then
A. $f \left( - \frac { 1 } { 2 } \right) = 0$
B. $f ( - 1 ) = 0$
C. $f ( 2 ) = 0$
D. $f ( 4 ) = 0$

Given that $f ( x ) = \frac { x e ^ { x } } { e ^ { a x } - 1 }$ is an even function, then $a =$
A. $- 2$
B. $- 1$
C. 1
D. 2

If a function $\mathrm { f } : [ - 2 \mathrm { a } , 2 \mathrm { a } ] - - > \mathrm { R }$ is an odd function such that $\mathrm { f } ( \mathrm { x } ) = \mathrm { f } ( 2 \mathrm { a } - \mathrm { x } )$ for $\mathrm { x } \hat { \mathrm { I } } [ \mathrm { a } , 2 \mathrm { a } [$ and the left hand derivative at $\mathrm { x } = \mathrm { a }$ is 0 then find the left hand derivative at $\mathrm { x } = - \mathrm { a }$.

A function f defined on the set of real numbers satisfies the inequality
$$f ( x ) < f ( x + 2 )$$
for every real number x.
Accordingly,
I. $f ( 1 ) < f ( 5 )$ II. $| f ( - 1 ) | < | f ( 1 ) |$ III. $f ( 0 ) + f ( 2 ) < 2 \cdot f ( 4 )$
Which of these statements are always true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

Function f is defined for every $\mathrm { x } \in ( 0,3 ]$ as
$$f ( x ) = 2 x + 1$$
and satisfies the equality
$$f ( x ) = f ( x + 3 )$$
for every real number x. Accordingly, what is the sum $\mathbf { f } ( \mathbf { 6 } ) + \mathbf { f } ( \mathbf { 7 } ) + \mathbf { f } ( \mathbf { 8 } )$?
A) 8
B) 12
C) 15
D) 18
E) 21