Let $f(x) = xe^x$. Prove that the mapping $f$ establishes a bijection from the interval $] - \infty , - 1 ]$ onto the interval $\left[ - \mathrm { e } ^ { - 1 } , 0 [ \right.$. In the rest of the problem, the inverse of this bijection is denoted $V$.

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 $I$ 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 every $x \in I$. 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 $x$.

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 $x$.

Prove that the endomorphism $D - I$ is invertible and express $L$ in terms of $(D-I)^{-1}$, where $L$ is defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}^{2}$ be a function given by $f(x) = (x^{m}, x^{n})$, where $x \in \mathbb{R}$ and $m, n$ are fixed positive integers. Suppose that $f$ is one-one. Then
(a) Both $n$ and $m$ must be odd
(b) At least one of $m$ and $n$ must be odd
(c) Exactly one of $m$ and $n$ must be odd
(d) Neither $m$ nor $n$ can be odd.

Let $f(x) = \dfrac{x^2}{x-1}$. Which of the following is true?
(A) $f$ is neither one-one nor onto
(B) $f$ is one-one and onto
(C) $f$ is one-one but not onto
(D) $f$ is onto but not one-one

Let $\mathbb{R}$ be the set of all real numbers. The function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^3 - 3x^2 + 6x - 5$ is
(A) one-to-one, but not onto
(B) one-to-one and onto
(C) onto, but not one-to-one
(D) neither one-to-one nor onto.

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) $x = 0, 4$

Let $\mathbb{R}$ be the set of all real numbers. The function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^3 - 3x^2 + 6x - 5$ is
(A) one-to-one, but not onto
(B) one-to-one and onto
(C) onto, but not one-to-one
(D) neither one-to-one nor onto

Let $\mathbb { R }$ be the set of all real numbers. The function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 5$ is
(A) one-to-one, but not onto
(B) one-to-one and onto
(C) onto, but not one-to-one
(D) neither one-to-one nor onto

Let $f : [-1,1] \rightarrow \mathbb{R}$ be a function such that $f\left(\sin\frac{x}{2}\right) = \sin x + \cos x$, for all $x \in [-\pi, \pi]$. The value of $f\left(\frac{3}{5}\right)$ is
(A) $\frac{24}{25}$
(B) $\frac{31}{25}$
(C) $\frac{33}{25}$
(D) $\frac{7}{5}$.

For every real number $x \neq - 1$, let $f ( x ) = \frac { x } { x + 1 }$. Write $f _ { 1 } ( x ) = f ( x )$ and for $n \geq 2 , f _ { n } ( x ) = f \left( f _ { n - 1 } ( x ) \right)$. Then,
$$f _ { 1 } ( - 2 ) \cdot f _ { 2 } ( - 2 ) \cdots \cdots f _ { n } ( - 2 )$$
must equal
(A) $\frac { 2 ^ { n } } { 1 \cdot 3 \cdot 5 \cdots \cdot ( 2 n - 1 ) }$
(B) 1
(C) $\frac { 1 } { 2 } \binom { 2 n } { n }$
(D) $\binom { 2 n } { n }$.

Consider the function $f : \mathbb { C } \rightarrow \mathbb { C }$ defined by $$f ( a + i b ) = e ^ { a } ( \cos b + i \sin b ) , a , b \in \mathbb { R }$$ where $i$ is a square root of $-1$. Then
(A) $f$ is one-to-one and onto.
(B) $f$ is one-to-one but not onto.
(C) $f$ is onto but not one-to-one.
(D) $f$ is neither one-to-one nor onto.

Let $A = \{1, \ldots, 5\}$ and $B = \{1, \ldots, 10\}$. Then the number of ordered pairs $(f, g)$ of functions $f : A \rightarrow B$ and $g : B \rightarrow A$ satisfying $(g \circ f)(a) = a$ for all $a \in A$ is
(A) $\frac{10!}{5!} \times 5^5$
(B) $5^{10} \times 5!$
(C) $10! \times 5!$
(D) $\binom{10}{5} \times 10^5$

Consider the following two statements: (I) There exists a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g\left(x^3 + x^5\right) = e^x - 100$. (II) There exists a continuous function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g\left(e^x\right) = x^3 + x^5$. Then
(A) Only (I) is correct.
(B) Only (II) is correct.
(C) Both (I) and (II) are correct.
(D) Neither (I) nor (II) is correct.

Let $\mathbb { R }$ be the set of all real numbers. The function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 5$ is
(a) one-to-one, but not onto.
(B) one-to-one and onto.
(C) onto, but not one-to-one.
(D) neither one-to-one nor onto.

Let $f: \{1,2,3,4\} \to \{1,2,3,4\}$ and $g: \{1,2,3,4\} \to \{1,2,3,4\}$ be invertible functions such that $f \circ g = $ identity. Then
(A) $f = g^{-1}$
(B) $g = f^{-1}$
(C) $f \circ g \neq g \circ f$
(D) $f \circ g = g \circ f$

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

If the function $f(x)=x^{3}+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is

Let $f$ be a real-valued function defined on the interval $( - 1,1 )$ such that $e ^ { - x } f ( x ) = 2 + \int _ { 0 } ^ { x } \sqrt { t ^ { 4 } + 1 } d t$, for all $x \in ( - 1,1 )$, and let $f ^ { - 1 }$ be the inverse function of $f$. Then $\left( f ^ { - 1 } \right) ^ { \prime } ( 2 )$ is equal to
A) 1
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 1 } { e }$

Let $f ( x ) = x ^ { 2 }$ and $g ( x ) = \sin x$ for all $x \in \mathbb { R }$. Then the set of all $x$ satisfying $( f \circ g \circ g \circ f ) ( x ) = ( g \circ g \circ f ) ( x )$, where $( f \circ g ) ( x ) = f ( g ( x ) )$, is
(A) $\pm \sqrt { n \pi } , n \in \{ 0,1,2 , \ldots \}$
(B) $\pm \sqrt { n \pi } , n \in \{ 1,2 , \ldots \}$
(C) $\frac { \pi } { 2 } + 2 n \pi , n \in \{ \ldots , - 2 , - 1,0,1,2 , \ldots \}$
(D) $2 n \pi , n \in \{ \ldots , - 2 , - 1,0,1,2 , \ldots \}$

Let $f:\mathbb{R} \rightarrow \mathbb{R}, g:\mathbb{R} \rightarrow \mathbb{R}$ and $h:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that $f(x) = x^3 + 3x + 2, g(f(x)) = x$ and $h(g(g(x))) = x$ for all $x \in \mathbb{R}$. Then
(A) $g'(2) = \frac{1}{15}$
(B) $h'(1) = 666$
(C) $h(0) = 16$
(D) $h(g(3)) = 36$

Let $E _ { 1 } = \left\{ x \in \mathbb { R } : x \neq 1 \right.$ and $\left. \frac { x } { x - 1 } > 0 \right\}$ and $E _ { 2 } = \left\{ x \in E _ { 1 } : \sin ^ { - 1 } \left( \log _ { e } \left( \frac { x } { x - 1 } \right) \right) \right.$ is a real number $\}$. (Here, the inverse trigonometric function $\sin ^ { - 1 } x$ assumes values in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$.) Let $f : E _ { 1 } \rightarrow \mathbb { R }$ be the function defined by $f ( x ) = \log _ { e } \left( \frac { x } { x - 1 } \right)$ and $g : E _ { 2 } \rightarrow \mathbb { R }$ be the function defined by $g ( x ) = \sin ^ { - 1 } \left( \log _ { e } \left( \frac { x } { x - 1 } \right) \right)$.
LIST-I P. The range of $f$ is Q. The range of $g$ contains R. The domain of $f$ contains S. The domain of $g$ is
LIST-II

1. $\left( - \infty , \frac { 1 } { 1 - e } \right] \cup \left[ \frac { e } { e - 1 } , \infty \right)$
2. $( 0,1 )$
3. $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
4. $( - \infty , 0 ) \cup ( 0 , \infty )$
5. $\left( - \infty , \frac { e } { e - 1 } \right]$
6. $( - \infty , 0 ) \cup \left( \frac { 1 } { 2 } , \frac { e } { e - 1 } \right]$

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 1 }$
(B) $\mathbf { P } \rightarrow \mathbf { 3 } ; \mathbf { Q } \rightarrow \mathbf { 3 } ; \mathbf { R } \rightarrow \mathbf { 6 } ; \mathbf { S } \rightarrow \mathbf { 5 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 6 }$
(D) $\mathrm { P } \rightarrow 4 ; \mathrm { Q } \rightarrow 3 ; \mathrm { R } \rightarrow 6 ; \mathrm { S } \rightarrow 5$

If the function $f : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $f ( x ) = | x | ( x - \sin x )$, then which of the following statements is TRUE?
(A) $f$ is one-one, but NOT onto
(B) $f$ is onto, but NOT one-one
(C) $f$ is BOTH one-one and onto
(D) $f$ is NEITHER one-one NOR onto