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

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

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]

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

Given $f ( x ) = x ^ { 3 }$, find $f ^ { -1 } ( x ) =$ $\_\_\_\_$

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. Determine an equivalent of $W ( x )$ as $x \rightarrow 0$ as well as an equivalent of $W ( x )$ as $x \rightarrow + \infty$.

Let $f(x) = xe^x$, and let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively. For a real parameter $m$, we consider the equation with unknown $x \in \mathbb { R }$
$$x \mathrm { e } ^ { x } = m \tag{I.1}$$
Determine, as a function of $m$, the number of solutions of (I.1). Explicitly express the possible solutions using the functions $V$ and $W$.

Let $f(x) = \dfrac{1}{x-2}$. Find the $x$-coordinates of the points where $f(x) = f^{-1}(x)$.
(A) $x = 1 \pm \sqrt{2}$ (B) $x = 2 \pm \sqrt{2}$ (C) $x = 1 \pm \sqrt{3}$ (D) $x = 0, 4$

The number of solutions of the equation $\sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right) = \sin^{-1}\left(\frac{x-1}{\sqrt{1+(x-1)^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+(x-1)^2}}\right)$ is
(A) 0
(B) 1
(C) 2
(D) infinite

The function $f : R \sim \{ 0 \} \rightarrow R$ given by $f ( x ) = \frac { 1 } { x } - \frac { 2 } { e ^ { 2 x } - 1 }$ can be made continuous at $x = 0$ by defining $f ( 0 )$ as
(1) 2
(2) - 1
(3) 0
(4) 1

If the function $f ( x ) = \left\{ \begin{array} { c l } \frac { \sqrt { 2 + \cos x } - 1 } { ( \pi - x ) ^ { 2 } } , & x \neq \pi \\ k , & x = \pi \end{array} \right.$ is continuous at $x = \pi$, then $k$ equals
(1) $\frac { 1 } { 4 }$
(2) 0
(3) 2
(4) $\frac { 1 } { 2 }$

If the function $f$ defined on $\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)$ by $f ( x ) = \left\{ \begin{array} { l l } \frac { \sqrt { 2 } \cos x - 1 } { \cot x - 1 } , & x \neq \frac { \pi } { 4 } \\ k , & x = \frac { \pi } { 4 } \end{array} \right.$ is continuous, then $k$ is equal to
(1) $\frac { 1 } { 2 }$
(2) 1
(3) 2
(4) $\frac { 1 } { \sqrt { 2 } }$

If $R = \{(x, y) : x, y \in Z, x^{2} + 3y^{2} \leq 8\}$ is a relation on the set of integers $Z$, then the domain of $R^{-1}$ is
(1) $\{-2, -1, 1, 2\}$
(2) $\{0, 1\}$
(3) $\{-2, -1, 0, 1, 2\}$
(4) $\{-1, 0, 1\}$

The number of real roots of the equation $\tan ^ { - 1 } \sqrt { x ( x + 1 ) } + \sin ^ { - 1 } \sqrt { x ^ { 2 } + x + 1 } = \frac { \pi } { 4 }$ is:
(1) 1
(2) 2
(3) 4
(4) 0

If for $p \neq q \neq 0$, then function $f ( x ) = \frac { \sqrt [ 7 ] { p ( 729 + x ) } - 3 } { \sqrt [ 3 ] { 729 + q x } - 9 }$ is continuous at $x = 0$, then
(1) $7 p q f ( 0 ) - 1 = 0$
(2) $63 q f ( 0 ) - p ^ { 2 } = 0$
(3) $21 q f ( 0 ) - p ^ { 2 } = 0$
(4) $7 p q f ( 0 ) - 9 = 0$

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