For the function $y = \sqrt { 4 - 2 x } + 3$, what is the minimum value of the real number $k$ such that the graph of its inverse function and the line $y = - x + k$ intersect at two distinct points? [3 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

For the function $f ( x ) = \left( x ^ { 2 } + 2 \right) e ^ { - x }$, the function $g ( x )$ is differentiable and satisfies
$$g \left( \frac { x + 8 } { 10 } \right) = f ^ { - 1 } ( x ) , \quad g ( 1 ) = 0$$
Find the value of $\left| g ^ { \prime } ( 1 ) \right|$. [4 points]

For the set $X = \{ 1,2,3,4 \}$, how many functions $f : X \rightarrow X$ satisfy the following condition? [3 points] $\square$
(1) 64
(2) 68
(3) 72
(4) 76
(5) 80

Consider the function $$f ( x ) = \begin{cases} - 3 x + a & ( x \leq 1 ) \\ \frac { x + b } { \sqrt { x + 3 } - 2 } & ( x > 1 ) \end{cases}$$ If $f ( x )$ is continuous on the entire set of real numbers, find the value of $a + b$. (Here, $a$ and $b$ are constants.) [4 points]

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

The function $$f(x) = \left\{ \begin{array}{cc} 5x + a & (x < -2) \\ x^{2} - a & (x \geq -2) \end{array} \right.$$ is continuous on the set of all real numbers. What is the value of the constant $a$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For a cubic function $f(x)$ with leading coefficient 1, let the function $g(x)$ be $$g(x) = f\left(e^{x}\right) + e^{x}$$ The tangent line to the curve $y = g(x)$ at the point $(0, g(0))$ is the $x$-axis, and the function $g(x)$ has an inverse function $h(x)$. What is the value of $h'(8)$? [3 points]
(1) $\frac{1}{36}$
(2) $\frac{1}{18}$
(3) $\frac{1}{12}$
(4) $\frac{1}{9}$
(5) $\frac{5}{36}$

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

6. The domain of the function $f ( x ) = \sqrt { 4 - | x | } + \lg \frac { x ^ { 2 } - 5 x + 6 } { x - 3 }$ is
A. $ ( 2,3 )$
B. $ ( 2,4 ]$
C. $ ( 2,3 ) \cup ( 3,4 ]$
D. $ ( - 1,3 ) \cup ( 3,6 ]$

7. There exists a function $f ( x )$ satisfying, for all $x \in \mathbb{R}$,
A. $f ( \sin 2 x ) = \sin x$
B. $f ( \sin 2 x ) = x ^ { 2 } + x$
C. $f \left( x ^ { 2 } + 1 \right) = | x + 1 |$
D. $f \left( x ^ { 2 } + 2 x \right) = | x + 1 |$

It is known that the function $f(x)$ is an odd function defined on $\mathbb{R}$. When $x \in (-\infty, 0)$, $f(x) = 2x^3 + x^2$. Then $f(2) = $ \_\_\_\_

Let $f ( x ) = \begin{cases} 2 ^ { -x } & x \leq 0 \\ 1 & x > 0 \end{cases}$. Then the range of $x$ satisfying $f ( x + 1 ) < f ( 2 x )$ is
A. $( - \infty , - 1 ]$
B. $( 0 , + \infty )$
C. $( - 1,0 )$
D. $( - \infty , 0 )$

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

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. $f ( 4 ) = 0$

If $f ( x ) = \ln \left| a + \frac { 1 } { 1 - x } \right| + b$ is an odd function, then $a = $ $\_\_\_\_$ . $b = $ $\_\_\_\_$ .

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

We are given a function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ and we define a function $f_\xi : \mathcal{M}_d(\mathbb{R}) \rightarrow \mathcal{M}_d(\mathbb{R})$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad f_\xi(A) = \left(\xi\left(A_{i,j}\right)\right)_{1 \leqslant i,j \leqslant d}$$ We propose to determine the continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ such that $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) \text{ invertible} \tag{V.1}$$
Determine the continuous functions $\xi$ satisfying condition (V.1) when $d = 1$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Deduce that the function $\xi$ is injective, then that it is strictly monotone on $\mathbb{R}$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Show that the function $\xi$ does not vanish on $\mathbb{R}^*$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
The purpose of this question is to show $\xi(0) = 0$.
1) Show that if $\xi(0) \neq 0$, then there exists $\alpha > 0$ such that $\xi(0)\xi(2) = \xi(1)\xi(\alpha)$.
2) Conclude.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Let $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ be the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$. Show that where it is defined $$(\eta(xy))^2 = \eta\left(x^2\right)\eta\left(y^2\right)$$

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), and $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ is the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$.
We assume in this question that the function $\eta$ takes strictly positive values on $I \cap {]0, +\infty[}$.
1) Show that the function $f = \ln \circ \eta \circ \exp$ satisfies equation (IV.1) on an interval $]-\infty, M[$, with $M$ (possibly infinite) to be determined as a function of the interval $I$.
2) Deduce that on the interval $I \cap {]0, +\infty[}$ the function $\eta$ is of the form $$\eta : x \mapsto K_1 x^{\alpha_1}$$ with two constants $K_1 > 0$ and $\alpha_1 > 0$.
3) Show that on the interval $I \cap {]-\infty, 0[}$ the function $\eta$ is of the form $$\eta : x \mapsto K_2(-x)^{\alpha_2}$$ with two constants $K_2 < 0$ and $\alpha_2 > 0$.
4) Show that $I = \mathbb{R}$ then that the function $\eta$ is an odd function.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Deduce in the general case that, if $\xi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function satisfying condition (V.1), then it is odd and its restriction to $\mathbb{R}_+^*$ is of the form $x \mapsto Cx^\beta$, with $C \neq 0$ and $\beta > 0$.

Justify that the mapping $f$ establishes a bijection from the interval $\left[ - 1 , + \infty \left[ \right. \right.$ onto the interval $\left[ - e ^ { - 1 } , + \infty [ \right.$, where $f : \mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto x\mathrm{e}^{x}$.

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 $x \rightarrow + \infty$.