If the function $f ( x ) = \left\{ \begin{array} { c l } ( 1 + | \cos x | ) \frac { \lambda } { | \cos x | } , & 0 < x < \frac { \pi } { 2 } \\ \mu , & x = \frac { \pi } { 2 } \\ e ^ { \frac { \cot 6 x } { \cot 4 x } } , & \frac { \pi } { 2 } < x < \pi \end{array} \right.$ is continuous at $x = \frac { \pi } { 2 }$, then $9 \lambda + 6 \log _ { e } \mu + \mu ^ { 6 } - e ^ { 6 \lambda }$ is equal to
(1) 11
(2) 8
(3) $2 e ^ { 4 } + 8$
(4) 10

If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { 72 ^ { x } - 9 ^ { x } - 8 ^ { x } + 1 } { \sqrt { 2 } - \sqrt { 1 + \cos x } } , & x \neq 0 \\ a \log _ { e } 2 \log _ { e } 3 & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then the value of $a ^ { 2 }$ is equal to
(1) 968
(2) 1152
(3) 746
(4) 1250

$$f ( x ) = \arcsin \left( \frac { x } { 3 } + 2 \right)$$
Which of the following is the inverse function $\mathbf { f } ^ { \mathbf { - 1 } } ( \mathbf { x } )$ of this function?
A) $2 \sin ( x ) - 6$
B) $2 \sin ( x ) + 3$
C) $3 \sin ( x ) - 6$
D) $\sin ( 2 x - 6 )$
E) $\sin ( 2 x ) - 3$

The following functions are given:
$f(x) = 3x - 6$
$g(x) = (x - 2)^{2}$
Accordingly, $\left(g \circ f^{-1}\right)(x)$ is equal to which of the following?
A) $\frac{3x^{2}}{2} - 1$ B) $(3x + 4)^{2}$ C) $x^{2} - 4x + 2$ D) $\frac{x^{2}}{9}$ E) $(3x - 8)^{2}$

$f : [ 1 , \infty ) \rightarrow [ 1 , \infty )$ is a function and
$$f \left( e ^ { x } \right) = \sqrt { x } + 1$$
Given this, what is the value of $f ^ { - 1 } ( 2 )$?
A) 1
B) $e - 1$
C) e
D) $e ^ { 2 }$
E) $\ln 2$

$$\begin{aligned} & f ( x ) = - \log _ { 2 } x \\ & g ( x ) = \log _ { 10 } x \end{aligned}$$
Given this, what is the value of a that satisfies the equality $\left( \right.$ gof $\left. ^ { - 1 } \right) ( a ) = \ln 2$?
A) $\ln 2$
B) $\frac { \ln 2 } { \ln 10 }$
C) $\frac { \ln 10 } { \ln 2 }$
D) $\ln \left( \frac { 1 } { 10 } \right)$
E) $\ln \left( \frac { 1 } { 2 } \right)$

For the function $f ( x ) = \log _ { x } 2$,
$$f \left( 4 ^ { a } \right) \cdot f ^ { - 1 } \left( \frac { 1 } { 3 } \right) = 6$$
What is the value of a that satisfies this equation?
A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 4 } { 3 }$

Let $a$ and $b$ be real numbers. A function $f$ that is continuous on the set of real numbers is defined as
$$f ( x ) = \begin{cases} x ^ { 2 } - 4 & , x \leq a \\ 5 x - 8 & , a < x \leq b \\ 7 & , x > b \end{cases}$$
Accordingly, what is the sum $a + b$?
A) 4
B) 5
C) 6
D) 7
E) 8

The graph of a function $f$ in the rectangular coordinate plane is given below.
A function $g$ defined on the set of real numbers has a limit at all points where it is defined, and $\lim_{x \rightarrow 3} g(x) = 14$ is calculated.
If the function $f \cdot g$ is continuous on the set of real numbers, what is the value of $g(3)$?
A) 4 B) 6 C) 8 D) 10

Let $a$ be a non-zero real number, and $b$ and $c$ be real numbers. For the function $f(x) = ax + b$ defined on the set of real numbers and its inverse function $f^{-1}$,
$$\begin{aligned} & \lim_{x \rightarrow b} \frac{f(x)}{f^{-1}(x)} = c \\ & f(1) = 3 \end{aligned}$$
are given. Accordingly, what is the sum of the different values that $c$ can take?
A) 6 B) 7 C) 10 D) 11 E) 14