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

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

$$f(x) = \frac{\left(1+x+x^{2}+x^{3}\right)(1-x)^{2}}{1-x-x^{2}+x^{3}}$$
Given this, what is the value of $f(\sqrt{2})$?
A) 1
B) 2
C) 3
D) 4
E) 5

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) $\frac { 3 } { 2 }$

$$\begin{aligned} & f ( x ) = | 2 x - 5 | \\ & g ( x ) = | x + 1 | \end{aligned}$$
The functions are given. Accordingly, what is the sum of the x values that satisfy the equation $( g \circ f ) ( x ) = 3$?
A) $-3$
B) $-1$
C) 0
D) 2
E) 5

$$f ( x ) = - 3 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$$
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) $5 x ^ { 3 } - x ^ { 2 } + 2 x - 3$

The graph of the function $f : R \rightarrow R$ is given below.
Using the function f, the function g is defined for every $\mathrm { x } _ { 0 } \in \mathrm { R }$ as
$$g \left( x _ { 0 } \right) = f \left( x _ { 0 } \right) + \lim _ { x \rightarrow x _ { 0 } + } f ( x )$$
Accordingly, what is the value of (gof)(2)?
A) - 2
B) - 1
C) 0
D) 1
E) 2

Functions $f$ and $g$ defined on the set of real numbers satisfy the equalities
$$\begin{aligned} & ( f + g ) ( x ) = x ^ { 2 } \\ & ( f - g ) ( 2 x ) = x \end{aligned}$$
Accordingly, what is the product $f ( 4 ) \cdot g ( 4 )$?
A) 45
B) 51
C) 54
D) 60
E) 63

$$f ( x ) = \left\{ \begin{array} { c c } \frac { a x } { x + 2 b } \cdot \cot x & , x \neq 0 \\ 2 & , x = 0 \end{array} \right.$$
The function is continuous at the point $x = 0$. Accordingly, what is the ratio $\frac { a } { b }$?
A) 1
B) 2
C) 4
D) $\frac { 1 } { 3 }$
E) $\frac { 1 } { 6 }$

Functions $f$ and $g$ with domain of integers are defined as
$$\begin{aligned} & f ( n ) = n + \frac { 1 } { 3 } \\ & g ( n ) = n + \frac { 1 } { 6 } \end{aligned}$$
Given this, I. $f \circ f \circ f$ II. $f \circ g \circ f$ III. $g \circ f \circ g$ For which of these functions does the image set consist only of integers?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

Sets $A$, $B$, and $C$ are defined as $$\begin{aligned}& A = \{ ( x , x ) : x \in \mathbb { R } \} \\& B = \{ ( x , 3 - x ) : x \in \mathbb { R } \} \\& C = \{ ( x , x + 4 ) : x \in \mathbb { R } \}\end{aligned}$$ Given that $( p , q ) \in A \cap B$ and $( r , s ) \in B \cap C$, $$\frac { p - r } { q + s }$$ what is the value of this expression?\ A) $\frac { 1 } { 3 }$\ B) $\frac { 1 } { 4 }$\ C) $\frac { 3 } { 4 }$\ D) $\frac { 4 } { 5 }$\ E) $\frac { 2 } { 5 }$

Functions $f$ and $g$ are defined on the set of real numbers as $$\begin{aligned}& f ( x ) = \frac { x \cdot ( x - 2 ) } { 2 } \\& g ( x ) = \frac { x \cdot ( x - 1 ) \cdot ( x - 2 ) } { 3 }\end{aligned}$$ The sum of the $\mathbf{x}$ values satisfying the equality $$f ( 2 x ) = g ( x + 1 )$$ is what?\ A) 1\ B) 3\ C) 4\ D) 6\ E) 8

Let k be a real number. The functions f and g defined on the set of positive real numbers are
$$\begin{aligned} & f ( x ) = k x ^ { 2 } + 1 \\ & g ( x ) = \sqrt { x } + 2 \end{aligned}$$
defined in the form.
$$( f \circ g ) ( 9 ) = 6$$
Given that, what is the value of f(2)?
A) $\frac { 7 } { 5 }$ B) $\frac { 8 } { 5 }$ C) $\frac { 9 } { 5 }$ D) 2 E) 3

Let $a$ and $b$ be real numbers. The functions f and g are defined on the set of real numbers as
$$\begin{aligned} & f(x) = ax - b \\ & g(x) = bx - 2 \end{aligned}$$
Given that
$$\begin{aligned} & (f + g)(1) = f(1) \\ & (f + g)(2) = g(2) \end{aligned}$$
what is the product $\mathbf{a} \cdot \mathbf{b}$?
A) 2
B) 4
C) 6
D) 8
E) 10

In the rectangular coordinate plane, the graph of the function $f(x)$ defined on the closed interval $[0,5]$ is given in the figure.
If the function $(f \circ f \circ f)(x)$ attains its maximum value at the point $x = a$, in which of the following open intervals is the number $a$?
A) $( 0,1 )$
B) $( 1,2 )$
C) $( 2,3 )$
D) $( 3,4 )$
E) $( 4,5 )$

Let $a$ be a positive real number. The functions f and g are defined on the set of real numbers as
$$\begin{aligned} & f(x) = x + a \\ & g(x) = ax + 1 \end{aligned}$$
Given that $(\mathbf{f} \cdot \mathbf{g})(\mathbf{1}) = (\mathbf{f} + \mathbf{g})(\mathbf{2})$, what is $\mathbf{g}(\mathbf{7})$?
A) 8 B) 15 C) 22 D) 29 E) 36

Let $a$ and $b$ be real numbers. For functions $f$ and $g$ defined on the set of real numbers
$$\begin{aligned} & f(x) = x^{2} + ax + b \\ & g(x) = ax + 2 \\ & (f + g)(3) = 4 \\ & (f - g)(5) = 6 \end{aligned}$$
These equalities are satisfied.
Accordingly, what is the difference $\mathrm{a} - \mathrm{b}$?
A) 17 B) $\frac{52}{3}$ C) 18 D) $\frac{56}{3}$ E) 19

Functions $f$ and $g$ are defined on the set of real numbers as
$$\begin{aligned} & f(x) = \frac{3x + 4}{2} \\ & g(x) = \frac{2x - 4}{3} \end{aligned}$$
If $(\mathbf{f} \circ \mathbf{g})(\mathbf{a}) = \mathbf{f}(\mathbf{a}) = \mathbf{b}$, what is the product $\mathbf{a} \cdot \mathbf{b}$?
A) $-20$ B) $-12$ C) $-8$ D) 4 E) 16

Let $a$ and $b$ be real numbers. For the functions $f$ and $g$ defined on the set of real numbers as
$$\begin{aligned} & f(x) = \frac{x}{2} + 1 \\ & g(x) = 2x - 3 \end{aligned}$$
the equalities
$$\begin{aligned} & (f + g)(a) = f(a) \\ & (f - g)(b) = g(b) \end{aligned}$$
are satisfied. Accordingly, what is the value of $(f \circ g)(a \cdot b)$?
A) $\frac{1}{2}$ B) $\frac{5}{2}$ C) $\frac{9}{2}$ D) $\frac{13}{2}$ E) $\frac{17}{2}$