Q71. Let $f ( x ) = \left\{ \begin{array} { c c c } - \mathrm { a } & \text { if } & - \mathrm { a } \leq x \leq 0 \\ x + \mathrm { a } & \text { if } & 0 < x \leq \mathrm { a } \end{array} \right.$ where $\mathrm { a } > 0$ and $\mathrm { g } ( x ) = ( f ( x \mid ) - | f ( x ) | ) / 2$. Then the function $g : [ - a , a ] \rightarrow [ - a , a ]$ is
(1) neither one-one nor onto.
(2) onto.
(3) both one-one and onto.
(4) one-one.

Q72. The function $\mathrm { f } : \mathrm { R } - > \mathrm { R } , f ( x ) = \frac { x ^ { 2 } + 2 x - 15 } { x ^ { 2 } - 4 x + 9 } , x \in \mathbb { R }$ is
(1) one-one but not onto.
(2) both one-one and onto.
(3) onto but not one-one.
(4) neither one-one nor onto.

Q85. Consider the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \frac { 2 x } { \sqrt { 1 + 9 x ^ { 2 } } }$. If the composition of $f , \underbrace { ( f \circ f \circ f \circ \cdots \circ f ) } _ { 10 \text { times } } ( x ) = \frac { 2 ^ { 10 } x } { \sqrt { 1 + 9 \alpha x ^ { 2 } } }$, then the value of $\sqrt { 3 \alpha + 1 }$ is equal to $\_\_\_\_$

If the domain of the function $\frac { 1 } { \ln ( 10 - x ) } + \sin ^ { - 1 } \left( \frac { x + 2 } { 2 x + 3 } \right)$ is $( - \infty , - a ] \cup ( - 1 , b ) \cup ( b , c )$, then $( b + c - 3 a )$ is equal to
(A) $20 - \frac { 5 } { 3 }$
(B) $21$
(C) 23
(D) 24

Let $f$ be a function such that $3 f ( x ) + 2 f \left( \frac { m } { 19 x } \right) = 5 x , x \neq 0$ where $m = \sum _ { i = 1 } ^ { 9 } ( i ) ^ { 2 }$, then $f ( 5 ) - f ( 2 )$ is equal to

If $g ( x ) = 3 x ^ { 2 } + 2 x - 3 , f ( 0 ) = - 3, 4 g ( f ( x ) ) = 3 x ^ { 2 } - 32 x + 72$. Then $\mathrm { f } ( \mathrm { g } ( 2 ) )$ is equal to (A) $- \frac { 25 } { 6 }$ (B) $\frac { 25 } { 6 }$ (C) $\frac { 7 } { 2 }$ (D) $\frac { 5 } { 2 }$

The domain of $\operatorname { Sin } ^ { - 1 } \left( \frac { 1 } { x ^ { 2 } - 2 x - 1 } \right)$ is $( - \infty , \alpha ] \cup [ \beta , \gamma ] \cup [ \delta , \infty )$.
The value of $\alpha + \beta + \gamma + \delta$ is equal to:

2. For ALL APPLICANTS.

Let
$$f ( x ) = x + 1 \quad \text { and } \quad g ( x ) = 2 x$$
We will, for example, write $f g$ to denote the function "perform $g$ then perform $f$ " so that
$$f g ( x ) = f ( g ( x ) ) = 2 x + 1$$
If $i \geqslant 0$ is an integer we will, for example, write $f ^ { i }$ to denote the function which performs $f i$ times, so that
$$f ^ { i } ( x ) = \underbrace { f f f \cdots f } _ { i \text { times } } ( x ) = x + i .$$
(i) Show that
$$f ^ { 2 } g ( x ) = g f ( x )$$
(ii) Note that
$$g f ^ { 2 } g ( x ) = 4 x + 4$$
Find all the other ways of combining $f$ and $g$ that result in the function $4 x + 4$.
(iii) Let $i , j , k \geqslant 0$ be integers. Determine the function
$$f ^ { i } g f ^ { j } g f ^ { k } ( x )$$
(iv) Let $m \geqslant 0$ be an integer. How many different ways of combining the functions $f$ and $g$ are there that result in the function $4 x + 4 m$ ?

2. For ALL APPLICANTS.

(i) Let $k \neq \pm 1$. The function $f ( t )$ satisfies the identity
$$f ( t ) - k f ( 1 - t ) = t$$
for all values of $t$. By replacing $t$ with $1 - t$, determine $f ( t )$.
(ii) Consider the new identity
$$f ( t ) - f ( 1 - t ) = g ( t )$$
(a) Show that no function $f ( t )$ satisfies $( * )$ when $g ( t ) = t$.
(b) What condition must the function $g ( t )$ satisfy for there to be a solution $f ( t )$ to $( * )$ ?
(c) Find a solution $f ( t )$ to $( * )$ when $g ( t ) = ( 2 t - 1 ) ^ { 3 }$.

2. For ALL APPLICANTS.

Let
$$A ( x ) = 2 x + 1 , \quad B ( x ) = 3 x + 2 .$$
(i) Show that $A ( B ( x ) ) = B ( A ( x ) )$.
(ii) Let $n$ be a positive integer. Determine $A ^ { n } ( x )$ where
$$A ^ { n } ( x ) = \underbrace { A ( A ( A \cdots A } _ { n \text { times } } ( x ) \cdots )$$
Put your answer in the simplest form possible.
A function $F ( x ) = 108 x + c$ (where $c$ is a positive integer) is produced by repeatedly applying the functions $A ( x )$ and $B ( x )$ in some order.
(iii) In how many different orders can $A ( x )$ and $B ( x )$ be applied to produce $F ( x )$ ? Justify your answer.
(iv) What are the possible values of $c$ ? Justify your answer.
(v) Are there positive integers $m _ { 1 } , \ldots , m _ { k } , n _ { 1 } , \ldots , n _ { k }$ such that
$$A ^ { m _ { 1 } } B ^ { n _ { 1 } } ( x ) + A ^ { m _ { 2 } } B ^ { n _ { 2 } } ( x ) + \cdots + A ^ { m _ { k } } B ^ { n _ { k } } ( x ) = 214 x + 92 \quad \text { for all } x ?$$
Justify your answer.
If you require additional space please use the pages at the end of the booklet

$$f\left(\frac{x-1}{x+1}\right) = x^{2} - x + 2$$
Given this, what is the value of $f(3)$?
A) 5
B) 6
C) 7
D) 8
E) 11

$$\begin{aligned} & f ( x ) = x ^ { 2 } \\ & g ( x ) = 2 x - 1 \end{aligned}$$
For these functions, what is $\mathbf { g } ( \mathbf { f } ( \mathbf { 2 } ) )$?
A) 0
B) 3
C) 5
D) 7
E) 9

Let x be a real number with $| x | \leq 4$, and
$$2 x + 3 y = 1$$
What is the sum of the integer values of y that satisfy this equation?
A) - 1
B) 0
C) 1
D) 2
E) 3

The following functions are given:
$f(x) = 3x - 6$
$g(x) = (x - 2)^{2}$
Accordingly, $\left(g \circ f^{-1}\right)(x)$ is equal to which of the following?
A) $\frac{3x^{2}}{2} - 1$ B) $(3x + 4)^{2}$ C) $x^{2} - 4x + 2$ D) $\frac{x^{2}}{9}$ E) $(3x - 8)^{2}$

The following functions are defined on the set of real numbers:
I. $f(x) = 2x - 1$ II. $g(x) = x^{2} + 2$ III. $h(x) = x^{3}$
Which of these functions are one-to-one?
A) I and II B) Only I C) I, II and III D) I and III E) Only II

$$f ( x ) = \arcsin \left( \frac { x } { 3 } + 2 \right)$$
Which of the following is the inverse function $\mathbf { f } ^ { \mathbf { - 1 } } ( \mathbf { x } )$ of this function?
A) $2 \sin ( x ) - 6$
B) $2 \sin ( x ) + 3$
C) $3 \sin ( x ) - 6$
D) $\sin ( 2 x - 6 )$
E) $\sin ( 2 x ) - 3$

The piecewise function $f : R \rightarrow R$ is defined as $f ( x ) = \left\{ \begin{array} { c l } 3 x + 1 , & x \text { is rational } \\ x ^ { 2 } , & x \text { is irrational } \end{array} \right.$
Accordingly, which of the following is $( f \circ f ) \left( \frac { \sqrt { 2 } } { 2 } \right)$?
A) $3 \sqrt { 2 } + 2$
B) $\sqrt { 2 } + 2$
C) $\frac { 1 } { 4 }$
D) $\frac { 5 } { 2 }$
E) $\frac { 7 } { 2 }$

$$\begin{aligned} & f ( x ) = 2 x - 1 \\ & g ( x ) = \frac { x } { 2 } - \frac { 1 } { x } \end{aligned}$$
Given this, what is the value of $\lim _ { x \rightarrow 2 } \frac { f ( g ( x ) ) } { x - 2 }$?
A) 0
B) 1
C) 3
D) $\frac { 1 } { 2 }$
E) $\frac { 3 } { 2 }$

Let $Z$ be the set of integers. The function $f : Z \rightarrow Z$ is defined as
$$f ( x ) = \begin{cases} x - 1 , & \text{if } x < 0 \\ x + 1 , & \text{if } x \geq 0 \end{cases}$$
Accordingly,
I. f is one-to-one. II. f is onto. III. The range of f is $Z \backslash \{ 0 \}$.
Which of these statements are true?
A) Only I
B) Only II
C) Only III
D) I and II
E) I and III

$$f ( x ) = \begin{cases} 1 , & x \leq 1 \\ x ^ { 2 } + a x + b , & 1 < x < 3 \\ 5 , & x \geq 3 \end{cases}$$
If the function is continuous on the set of real numbers, what is the difference $a - b$?
A) $-4$
B) $-1$
C) 2
D) 3
E) 5

For functions f and g defined on the set of real numbers
$$\begin{aligned} & f ( g ( x ) ) = x ^ { 2 } + 4 x - 1 \\ & g ( x ) = x + a \\ & f ^ { \prime } ( 0 ) = 1 \end{aligned}$$
Given this, what is a?
A) $-2$
B) $\frac { -1 } { 4 }$
C) 1
D) $\frac { 3 } { 2 }$
E) 3

$$f ( 2 x + 5 ) = \tan \left( \frac { \pi } { 2 } x \right)$$
For the function $f$ given by the equality, what is the value $f ^ { -1 } ( 1 )$?
A) $\frac { \pi } { 2 }$
B) $\frac { \pi } { 4 }$
C) $\pi$
D) $2 \pi$
E) $3 \pi$

$$f ( x ) = - 3 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$$
Given this, what is the product $x ^ { 3 } \cdot f \left( \frac { 1 } { x } \right)$ equal to?
A) $x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3$
B) $x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$
C) $3 x ^ { 3 } - 5 x ^ { 2 } + 2 x - 1$
D) $3 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 1$
E) $5 x ^ { 3 } - x ^ { 2 } + 2 x - 3$

$f : [ 1 , \infty ) \rightarrow [ 1 , \infty )$ is a function and
$$f \left( e ^ { x } \right) = \sqrt { x } + 1$$
Given this, what is the value of $f ^ { - 1 } ( 2 )$?
A) 1
B) $e - 1$
C) e
D) $e ^ { 2 }$
E) $\ln 2$

I. $f ( x ) = 2 x$ II. $f ( x ) = 2 ^ { x }$ III. $f ( x ) = x ^ { 2 }$ Which of these functions satisfy the equation $f ( a + b ) = f ( a ) \cdot f ( b )$ for every real number a and b?
A) Only I
B) Only II
C) I and II
D) I and III
E) II and III