If $c$ is a real number with $0 < c < 1$, then show that the values taken by the function $y = \frac{x^2 + 2x + c}{x^2 + 4x + 3c}$, as $x$ varies over real numbers, range over all real numbers.

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

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

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) $x = 0, 4$

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

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) $\frac{7}{5}$.

If $f , g$ are real-valued differentiable functions on the real line $\mathbb { R }$ such that $f ( g ( x ) ) = x$ and $f ^ { \prime } ( x ) = 1 + ( f ( x ) ) ^ { 2 }$, then $g ^ { \prime } ( x )$ equals
(A) $\frac { 1 } { 1 + x ^ { 2 } }$
(B) $1 + x ^ { 2 }$
(C) $\frac { 1 } { 1 + x ^ { 4 } }$
(D) $1 + x ^ { 4 }$.

Let $\mathbb { R }$ be the set of all real numbers. The function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 5$ 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.

23. If $g ( f ( x ) ) = | \sin x |$ and $f ( g ( x ) ) = ( \sin \sqrt { } x ) 2$, then :
(A) $f ( x ) = \sin 2 x , g ( x ) = \sqrt { } x$
(B) $f ( x ) = \sin x , g ( x ) = | x |$
(C) $f ( x ) = x 2 , g ( x ) = \sin \sqrt { } x$
(D) $f$ and $g$ cannot be determined

28. If $f ( x ) = 3 x - 5$, then $f - 1 ( x )$
(A) is given by $1 / ( 3 x - 5 )$.
(B) is given by $( x + 5 ) / 3 \quad$.
(C) does not exist because $f$ is not one-one
(D) does not exist because $f$ is not onto.

15. The function $f ( x ) = [ x ] 2 - [ x 2 ]$ (where [y] is the greatest integer less than or equal to $y$ ), is discontinuous at :
(A) all integers
(B) all integers except 0 and 1
(C) all integers except 0
(D) all integers except 1

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) $- \infty < x < 1$

8. If $f : [ 1 , \infty )$ is given by $f ( x ) = x + 1 / x$ then $f - 1 ( x )$ equals :
(A) $( x + \sqrt { } ( x 2 - 4 ) ) / 2$
(B) $x / 1 + x 2$
(C) $( x - \sqrt { } ( x 2 - 4 ) ) / 2$
(D) $1 + \sqrt { } ( \times 2 - 4 )$

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) $( - 3 , \infty ) \backslash \{ - 1 , - 2 \}$

25. Let $f ( x ) = a x / ( x + 1 ) , x \neq - 1$. Then for what value of $a$ is $f [ f ( x ) ] = x$ :
(A) $\sqrt { } 2$
(B) $- \sqrt { } 2$
(C) 1
(D) - 1

19. Suppose $f ( x ) = ( x + 1 ) ^ { 2 }$ for $x \geq - 1$ If $g ( x )$ is the function whose graph is reflection of the graph of $f ( x )$ with respect to the line $y = x$ then $g ( x )$ equals
(A) $\quad - \sqrt { } x - 1 , x \geq 0$
(B) $\quad 1 / ( x + 1 ) ^ { 2 } , x > - 1$
(C) $\quad \sqrt { } ( x + 1 ) , x \geq - 1$
(D) $\quad \sqrt { } \mathrm { x } - 1 , \mathrm { x } \geq 0$

20. Let function $f : R \rightarrow R$ be defined by $f ( x ) = 2 x + \sin x$ for $x \in R$ Then $f$ is
(A) One-to-one and onto
(B) One-to-one but NOT onto
(C) Onto but NOT one-to-one
(D) Neither one-to-one nor onto

If a function $\mathrm { f } : [ - 2 \mathrm { a } , 2 \mathrm { a } ] - - > \mathrm { R }$ is an odd function such that $\mathrm { f } ( \mathrm { x } ) = \mathrm { f } ( 2 \mathrm { a } - \mathrm { x } )$ for $\mathrm { x } \hat { \mathrm { I } } [ \mathrm { a } , 2 \mathrm { a } [$ and the left hand derivative at $\mathrm { x } = \mathrm { a }$ is 0 then find the left hand derivative at $\mathrm { x } = - \mathrm { a }$.

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 :

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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) $\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)$

14. $f ( x ) = \left\{ \begin{array} { l } x , \quad \text { if } x \text { is rational } \\ 0 , \quad \text { if } x \text { is irrational } \end{array} \right.$ and
$$g ( x ) = \left\{ \begin{array} { l } 0 , \quad \text { if } x \text { is rational } \\ x , \quad \text { if } x \text { is irrational. } \end{array} \text { Then } \mathrm { f } - \mathrm { g } \right. \text { is: }$$
(a) one-one and into
(b) neither one-one nor onto
(c) many one and onto
(d) one-one and onto

28. If $X$ and $Y$ are two non-empty sets where $f : X - - > Y$ is function is defined such that $\mathrm { f } ( \mathrm { c } ) = \{ \mathrm { f } ( \mathrm { x } ) : \mathrm { x }$ ÎC $\}$ for C ÍX and $f ^ { - 1 } ( D ) = \{ x : f ( x )$ Î $D \}$ for $D$ Í $y$, for any A Í X and B Í Y then :
(a) $f ^ { - 1 } ( f ( A ) ) = A$
(b) $f ^ { - 1 } ( f ( A ) ) = A$ only if $f ( X ) = Y$
(c) $f \left( f ^ { - 1 } ( B ) \right) = B$ only if B Í $f ( x )$
(d) $f \left( f ^ { - 1 } ( B ) \right) = B$