Given that $f ( x )$ is an odd function with domain $( - \infty , + \infty )$ satisfying $f ( 1 - x ) = f ( 1 + x )$. If $f ( 1 ) = 2$, then $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 30 ) =$
A. $- 50$
B. $0$
C. $2$
D. $50$

6. Let $f ( x )$ be an odd function, and when $x \geq 0$, $f ( x ) = \mathrm { e } ^ { x } - 1$. Then when $x < 0$, $f ( x ) =$
A. $\mathrm { e } ^ { - x } - 1$
B. $\mathrm { e } ^ { - x } + 1$
C. $- \mathrm { e } ^ { - x } - 1$
D. $- \mathrm { e } ^ { - x } + 1$

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

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

(1) [4 marks] Given is the function $f$ defined on $\mathbb { R }$ with $f ( x ) = e ^ { 2 x + 1 }$. Show that $f$ is invertible and determine a term for the inverse function of $f$.
(2a) [3 marks] Given is the function $g : x \mapsto \left( x ^ { 2 } - 9 x \right) \cdot \sqrt { 2 - x }$ with maximal domain $D _ { g }$. State $D _ { g }$ and all zeros of $g$.
(2b) [3 marks] Given is the function $h : x \mapsto \ln \left( \frac { 1 } { x ^ { 2 } + 1 } \right)$ defined on $\mathbb { R }$. Justify that the range of $h$ is the interval ] $- \infty ; 0 ]$.
Consider the function $f$ defined on $\mathbb { R } ^ { + }$ with $f ( x ) = \frac { 1 } { \sqrt { x ^ { 3 } } }$. (3a) [2 marks] Show that the function $F$ defined on $\mathbb { R } ^ { + }$ with $F ( x ) = - \frac { 2 } { \sqrt { x } }$ is an antiderivative of $f$.
(3b) [3 marks] The graph of $f$ encloses an area with the x-axis and the lines with equations $x = 1$ and $x = b$ with $b > 1$. Determine the value of $b$ for which this area has content 1.
Given are the function $f$ defined on $\mathbb { R }$ with $f ( x ) = \frac { 1 } { 8 } x ^ { 3 }$ and the points $Q _ { a } ( a \mid f ( a ) )$ for $a \in \mathbb { R }$. The figure shows the graph of $f$ as well as the points $P ( 0 \mid 2 )$ and $Q _ { 2 }$. [Figure]
(4a) [2 marks] Calculate for $a \neq 0$ the slope $m _ { a }$ of the line through the points $P$ and $Q _ { a }$ as a function of $a$. (for verification: $m _ { a } = \frac { a ^ { 3 } - 16 } { 8 a }$ )
)$} The tangent to the graph of $f$ at the point $Q _ { a }$ is denoted by $t _ { a }$. Determine computationally the value of $a \in \mathbb { R }$ for which $t _ { a }$ passes through $P$.
Given is the function $f : x \mapsto \frac { 6 x } { x ^ { 2 } - 4 }$ defined on $\mathbb { R } \backslash \{ - 2 ; 2 \}$. The graph of $f$ is denoted by $G _ { f }$ and is symmetric with respect to the origin.

Show that $x \mapsto \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ is an even function of the variable $x \in \mathbb { R }$.

Let $z \in \mathbb{C}$. Determine the domain of definition $D_z$ of the function $Z \mapsto \dfrac{1}{1 - 2zZ + Z^2}$.

Justify that every function $h : I \rightarrow \mathbb{R}$ can be written uniquely in the form $h = p + i$ with $p : I \rightarrow \mathbb{R}$ an even function and $i : I \rightarrow \mathbb{R}$ an odd function.

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. Explicitly determine $W ( 0 )$ and $W ^ { \prime } ( 0 )$.

Let $I$ be an open interval of $\mathbb { R }$. We are given a function $f : I \rightarrow \mathbb { R }$ of class $\mathcal { C } ^ { 3 }$, such that $f ^ { \prime } ( x ) > 0$ for all $x \in I$. Show that $f$ is bijective from $I$ onto the open interval $f ( I )$.
We denote by $g : f ( I ) \rightarrow I$ its inverse function. Recall the value of $g ^ { \prime } ( f ( x ) )$. Express $g ^ { \prime \prime } ( f ( x ) )$ as a function of the successive derivatives of $f$ at $x$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Show that $G_0'$ establishes a continuous bijection from $[u_0; +\infty[$ to $\mathbb{R}_+$. Deduce that the function $u : h \longmapsto u_h$ is continuous on $\mathbb{R}_+$ and differentiable on $\mathbb{R}_+^*$.

102- The figure shows the graph of $y = f(x)$. The domain of $y = \sqrt{xf(x)}$ is which of the following?
[Figure: Graph of $f(x)$ showing a curve passing through points $-3$, $1$, $2$ on the x-axis, with minimum near $x=-4$]

• [(1)] $[0, 2]$
• [(2)] $[-3, 2]$
• [(3)] $[-4,-3] \cup [1, 2]$
• [(4)] $[-3,0] \cup [1, 2]$

107- If $f(x) = 2x + 3$ and $g(f(x)) = 8x^2 + 22x + 20$, what is the rule of $f \circ g$?

• [(1)] $2x^2 - 7x + 3$
• [(2)] $2x^2 - x + 7$
• [(3)] $4x^2 - 2x + 13$
• [(4)] $4x^2 - 4x + 11$

108- The function $f(x) = x^2 + 2x + 1$ with domain $(-\infty, +\infty)$ is assumed to be invertible. The graphs of $f$ and $f^{-1}$ intersect at how many points?

• [(1)] $1$
• [(2)] $2$
• [(3)] $3$
• [(4)] No intersection

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) $4$

108- If $g(x)=2x-3$ and $(f\circ g)(x)=4(x^2-4x+5)$, what is $f(x)$?
(1) $x^2-4x+3$ (2) $x^2-4x+5$ (3) $x^2-2x+5$ (4) $x^2-2x+3$

110- The figure below shows the graph of the function $y = \sin^{-1}(U(x))$. What is the rule $U(x)$?
[Figure: graph of $y = \sin^{-1}(U(x))$ with a point marked at $x = -1$ and $x = 3$]

p{6cm}} (2) $\dfrac{2}{1-x}$	(1) $\dfrac{2}{x-1}$
[18pt] (4) $\dfrac{1}{2-x}$	(3) $\dfrac{1}{x-2}$

111- What is the value of the expression $169\sin\!\left(2\cos^{-1}\!\left(-\dfrac{5}{13}\right)\right)$?

p{6cm}} (2) $60$	(1) $-120$
[6pt] (4) $120$	(3) $-60$

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) $2.4$

117. If $f(x) = \dfrac{1}{2}(x + \sqrt{x^2 + 4})$, then $f^{-1}(x) + f^{-1}\!\left(\dfrac{1}{x}\right)$ equals which of the following?
(4) zero (3) $x^2 - 1$ (2) $\dfrac{2}{x}$ (1) $2x$
%% Page 5

1-1. If $f(x) = 3 - e^x$, and $g(x) = \sqrt{x f^{-1}(x)}$, what is the domain of $g$?
(1) $[0, 2]$ (2) $[0, 3)$ (3) $[2, 3)$ (4) $[1, 3)$

1-2. For which values of $a$ does the second-degree equation $a = 0$, i.e., $x^2 - 2(a-2)x + 14 - a = 0$, have two positive roots?
(1) $-2 < a < 2$ (2) $2 < a < 5$ (3) $2 < a < 14$ (4) $5 < a < 14$

1-3. The function $f(x) = a + \log_2(bx - 4)$ passes through the two points $(6, 2)$ and $(10, 12)$. What is $a$?
(1) $3$ (2) $4$ (3) $5$ (4) $6$

1-4. The figure shows part of the graph of the function $y = \dfrac{1}{2} + 2\cos mx$. At the point $x = \dfrac{16\pi}{3}$, what is the value of the function?

[Figure: Graph of a cosine-type function with period $4\pi$ shown on the $x$-axis]
(1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{2}$ (3) $1$ (4) zero

1-5. The graphs of the two functions $y = 3^x + \dfrac{4}{3}$ and $y = \left(\dfrac{\sqrt{3}}{3}\right)^{2x}$ intersect at point A. What is the distance of point A from the point $(1, -1)$?
(1) $1$ (2) $\sqrt{2}$ (3) $2$ (4) $\sqrt{5}$

1-6. For which value of $m$ is the sum of both roots of the second-degree equation $0 = 2x^2 - (m+1)x + \dfrac{1}{8}$ equal to $2$?
(1) $3$ (2) $4$ (3) $5$ (4) $6$

1-7. If $f(x) = \dfrac{1+x^2}{1-x^2}$ and $g(x) = \sqrt{x - x^2}$, what is the domain of $g \circ f$?
(1) $[0, 1)$ (2) $\{0\}$ (3) $(-1, 1)$ (4) $\mathbb{R} - \{1, -1\}$

1-8. What is $\sin\!\left(\dfrac{\pi}{2} + \cos^{-1}\!\left(-\dfrac{\sqrt{3}}{2}\right)\right)$?
(1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{2}$ (3) $1$ (4) zero
%% Page 4

107. If $f = \{(1,2),(2,5),(3,4),(4,6)\}$ and $g = \{(2,3),(4,2),(5,6),(3,1)\}$, then $\dfrac{g}{\text{g} \circ f^{-1}}$ equals which of the following?
(1) $\{(4,2),(5,2)\}$ (2) $\{(4,3),(3,5)\}$ (3) $\{(5,2),(2,4)\}$ (4) $\{(3,5),(2,4)\}$

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) $5$

108. The function $f(x) = x^2\sqrt{x}$ is one-to-one on a domain. Which of the following is the inverse function on this domain?
(1) $-\sqrt{x^2}\ ,\ x \leq 0$ (2) $-\sqrt[3]{x}\ ,\ x \leq 0$ (3) $-\sqrt{x^2}\ ,\ x \geq 0$ (4) $-\sqrt[3]{x}\ ,\ x \geq 0$

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) $\dfrac{1}{2}$

15. $f$ is a homographic function, $g(x) = \dfrac{1}{f(x)}$, and $\displaystyle\lim_{x \to -\infty} \dfrac{f(x)}{g^{-1}(x)} = \lim_{x \to +\infty} \dfrac{g^{-1}(x)}{g(x)}$. What value can $\displaystyle\lim_{x \to 0^+} f^{-1}(x)$ be?
\[ \text{(1) zero} \qquad \text{(2) } \dfrac{1}{2} \qquad \text{(3) } 1 \qquad \text{(4) } 2 \]