3. What is the domain of the following real valued function?
$$f ( x ) = \log _ { 2 } \left( x ^ { 2 } - 5 x + 6 \right)$$
(a) $( - \infty , 2 )$
(b) $( 3 , \infty )$
(c) $( - \infty , 2 ) \cup ( 3 , \infty )$
(d) $( - \infty , \infty )$

For a quadratic function $f ( x )$ with leading coefficient 1 and the function $$g ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { \ln ( x + 1 ) } & ( x \neq 0 ) \\ 8 & ( x = 0 ) \end{array} \right.$$ when the function $f ( x ) g ( x )$ is continuous on the interval $( - 1 , \infty )$, what is the value of $f ( 3 )$? [3 points]
(1) 6
(2) 9
(3) 12
(4) 15
(5) 18

19. (Total Score: 14 points) Subquestion 1 is worth 6 points, Subquestion 2 is worth 8 points.
Let the domain of function $f ( x ) = \sqrt { 2 - \frac { x + 3 } { x + 1 } }$ be $A$, and the domain of $g ( x ) = \lg [ ( x - a - 1 ) ( 2 a - x ) ]$ (where $a < 1$) be $B$.
(1) Find $A$;
(2) If $B \subseteq A$, find the range of real number $a$.

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 ]$

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. $( - \infty , 0 )$

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

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
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6. The domain of definition of the function $y ( x )$ is given by the equation $2 x + 2 y = 2$ is :
(A) $0 < x \leq 1$
(B) $0 \leq x \leq 1$
(C) $- \infty < x \leq 0$
(D) $- \infty < x < 1$

9. The domain of definition of $f ( x ) = ( \log 2 ( x + 3 ) ) / ( x 2 + 3 x + 2 )$ is:
(A) $\mathrm { R } \backslash \{ - 1 , - 2 \}$
B) $( - 2 , \infty )$
(C) $\mathrm { R } / \{ - 1 , - 2 , - 3 \}$
(D) $( - 3 , \infty ) \backslash \{ - 1 , - 2 \}$

12. Range of the function $f ( x ) = \left( x ^ { 2 } + x + 2 \right) / \left( x ^ { 2 } + x + 1 \right) ; x \hat { I } R$ is:
(a) $( 1 , ¥ )$
(b) $( 1,11 / 7 )$
(c) $( 1,7 / 3 )$
(d) $( 1,7 / 5 )$

17. Domain of definition of the function $f ( x ) = \sqrt { } \left( \sin ^ { - 1 } ( 2 x ) + \pi / 6 \right)$ for real valued $x$, is :

III askllTians ||

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(a) $\left[ - \frac { 1 } { 4 } , \frac { 1 } { 2 } \right]$
(b) $\left[ - \frac { 4 } { 2 } , \frac { 2 } { 2 } \right]$
(c) $\left( - \frac { 1 } { 2 } , \frac { 1 } { 9 } \right)$
(d) $\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)$

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) $\mathrm { P } \rightarrow 4 ; \mathrm { Q } \rightarrow 3 ; \mathrm { R } \rightarrow 6 ; \mathrm { S } \rightarrow 5$

The largest interval lying in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ for which the function $\left[ f ( x ) = 4 ^ { - x ^ { 2 } } + \cos ^ { - 1 } \left( \frac { x } { 2 } - 1 \right) + \log ( \cos x ) \right]$ is defined, is
(1) $[ 0 , \pi ]$
(2) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
(3) $\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
(4) $\left[ 0 , \frac { \pi } { 2 } \right)$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x) = \frac{x}{1+x^2}$, $x \in \mathbb{R}$. Then the range of $f$ is:
(1) $\mathbb{R} - [-1, 1]$
(2) $(-1, 1) - \{0\}$
(3) $\left[-\frac{1}{2}, \frac{1}{2}\right]$
(4) $\left(-\frac{1}{2}, \frac{1}{2}\right)$

The domain of the definition of the function $f ( x ) = \frac { 1 } { 4 - x ^ { 2 } } + \log _ { 10 } \left( x ^ { 3 } - x \right)$ is:
(1) $( - 1,0 ) \cup ( 1,2 ) \cup ( 2 , \infty )$
(2) $( 1,2 ) \cup ( 2 , \infty )$
(3) $( - 2 , - 1 ) \cup ( - 1,0 ) \cup ( 2 , \infty )$
(4) $( - 1,0 ) \cup ( 1,2 ) \cup ( 3 , \infty )$

Let $f:(1,3) \rightarrow R$ be a function defined by $f(x) = \frac{x[x]}{1 + x^{2}}$, where $[x]$ denotes the greatest integer $\leq x$. Then the range of $f$ is
(1) $\left(\frac{2}{5}, \frac{3}{5}\right] \cup \left(\frac{3}{4}, \frac{4}{5}\right)$
(2) $\left(\frac{2}{5}, \frac{1}{2}\right) \cup \left(\frac{3}{5}, \frac{4}{5}\right]$
(3) $\left(\frac{2}{5}, \frac{4}{5}\right]$
(4) $\left(\frac{3}{5}, \frac{4}{5}\right)$

The domain of the function $f(x) = \sin^{-1}\left(\frac{|x| + 5}{x^{2} + 1}\right)$ is $(-\infty, -a] \cup [a, \infty)$, then $a$ is equal to
(1) $\frac{\sqrt{17}}{2}$
(2) $\frac{\sqrt{17} - 1}{2}$
(3) $\frac{1 + \sqrt{17}}{2}$
(4) $\frac{\sqrt{17}}{2} + 1$

Let $f ( x ) = \sin ^ { - 1 } x$ and $g ( x ) = \frac { x ^ { 2 } - x - 2 } { 2 x ^ { 2 } - x - 6 }$. If $g ( 2 ) = \lim _ { x \rightarrow 2 } g ( x )$, then the domain of the function $f o g$ is
(1) $( - \infty , - 1 ] \cup [ 2 , \infty )$
(2) $( - \infty , - 2 ] \cup \left[ - \frac { 3 } { 2 } , \infty \right)$
(3) $( - \infty , - 2 ] \cup \left[ - \frac { 4 } { 3 } , \infty \right)$
(4) $( - \infty , - 2 ] \cup [ - 1 , \infty )$

Let $f : R \rightarrow R$ be defined as $f ( x ) = \begin{cases} 2 \sin \left( - \frac { \pi x } { 2 } \right) , & \text { if } x < - 1 \\ \left| a x ^ { 2 } + x + b \right| , & \text { if } - 1 \leq x \leq 1 \\ \sin ( \pi x ) , & \text { if } x > 1 \end{cases}$ If $f ( x )$ is continuous on $R$, then $a + b$ equals :
(1) 1
(2) 3
(3) - 3
(4) - 1

Let $\alpha \in R$ be such that the function $f ( x ) = \left\{ \begin{array} { l l } \frac { \cos ^ { - 1 } \left( 1 - \{ x \} ^ { 2 } \right) \sin ^ { - 1 } ( 1 - \{ x \} ) } { \{ x \} - \{ x \} ^ { 3 } } , & x \neq 0 \\ \alpha , & x = 0 \end{array} \right.$ is continuous at $x = 0$, where $\{ x \} = x - [ x ] , [ x ]$ is the greatest integer less than or equal to $x$. Then :
(1) $\alpha = \frac { \pi } { \sqrt { 2 } }$
(2) $\alpha = 0$
(3) no such $\alpha$ exists
(4) $\alpha = \frac { \pi } { 4 }$

Let $[ x ]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $f ( x ) = \sqrt { \frac { | [ x ] | - 2 } { | [ x ] | - 3 } }$ is $( - \infty , a ) \cup [ b , c ) \cup [ 4 , \infty ) , a < b < c$, then the value of $a + b + c$ is:
(1) 8
(2) 1
(3) $- 2$
(4) $- 3$

Let a function $f : R \rightarrow R$ be defined as, $f ( x ) = \begin{cases} \sin x - e ^ { x } & \text { if } x \leq 0 \\ a + [ - x ] & \text { if } 0 < x < 1 \\ 2 x - b & \text { if } x \geq 1 \end{cases}$
Where $[ x ]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$, then ( $a + b$ ) is equal to:
(1) 4
(2) 3
(3) 2
(4) 5

Let the functions $f : R \rightarrow R$ and $g : R \rightarrow R$ be defined as : $f ( x ) = \left\{ \begin{array} { l l } x + 2 , & x < 0 \\ x ^ { 2 } , & x \geq 0 \end{array} \right.$ and $g ( x ) = \begin{cases} x ^ { 3 } , & x < 1 \\ 3 x - 2 , & x \geq 1 \end{cases}$ Then, the number of points in $R$ where $( f \circ g ) ( x )$ is NOT differentiable is equal to :
(1) 3
(2) 1
(3) 0
(4) 2

The domain of $f ( x ) = \frac { \cos ^ { - 1 } \left( \frac { x ^ { 2 } - 5 x + 6 } { x ^ { 2 } - 9 } \right) } { \log \left( x ^ { 2 } - 3 x + 2 \right) }$ is
(1) $x \in \left[ \frac { - 1 } { 2 } , 1 \right) \cup ( 2 , \infty ) - \{ 3 \}$
(2) $x \in \left[ \frac { - 1 } { 2 } , 1 \right] \cup ( 2 , \infty ) - \{ 3 \}$
(3) $x \in \left( \frac { - 1 } { 2 } , 1 \right) \cup [ 2 , \infty ) - \{ 3 \}$
(4) $x \in \left[ \frac { - 1 } { 2 } , 1 \right) \cup [ 2 , \infty ) - \{ 3 \}$

The domain of the function $\cos ^ { - 1 } \left( \frac { 2 \sin ^ { - 1 } \left( \frac { 1 } { 4 x ^ { 2 } - 1 } \right) } { \pi } \right)$ is
(1) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right] \cup \left[ \frac { 1 } { \sqrt { 2 } } , \infty \right) \cup \{ 0 \}$
(2) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right] \cup \left[ \frac { 1 } { \sqrt { 2 } } , \infty \right)$
(3) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right) \cup \left( \frac { 1 } { 2 } , \infty \right) \cup \{ 0 \}$
(4) $R - \left\{ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right\}$