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. $- \mathrm { e } ^ { - x } + 1$
(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 $x \in \mathbb { R }$.
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 $I$ 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 $x=-4$]
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?
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) $4$
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 $169\sin\!\left(2\cos^{-1}\!\left(-\dfrac{5}{13}\right)\right)$?
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?
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) $-\sqrt[3]{x}\ ,\ x \geq 0$