$$\begin{aligned} & f ( x ) = | 2 x - 5 | \\ & g ( x ) = | x + 1 | \end{aligned}$$
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

A function f defined on the set of real numbers satisfies the inequality
$$f ( x ) < f ( x + 2 )$$
for every real number x.
Accordingly,
I. $f ( 1 ) < f ( 5 )$ II. $| f ( - 1 ) | < | f ( 1 ) |$ III. $f ( 0 ) + f ( 2 ) < 2 \cdot f ( 4 )$
Which of these statements are always true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

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

$$f ( x ) = \frac { - k x ^ { 3 } + k ^ { 2 } x } { k ^ { 3 } x ^ { 2 } + x - ( k + 1 ) }$$
The function has a vertical asymptote at $x = 1$. Accordingly, what is the value of $f ( 2 )$?
A) - 5
B) - 4
C) - 3
D) - 2
E) - 1

$$\lim _ { x \rightarrow 0 ^ { + } } ( \sin x ) \cdot ( \ln x )$$
Which of the following is this limit equal to?
A) $- 1$
B) 0
C) 1
D) $\infty$
E) $- \infty$

The function
$$f ( x ) = \frac { a x } { | b x + 2 | }$$
defined on a subset of the set of positive real numbers has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 4$.
Accordingly, what is the sum $a + b$?
A) 1
B) 2
C) 3
D) 4
E) 5

Below is the graph of the function $f$.
Accordingly, regarding the function f: I. The function f does not have an absolute maximum value on the interval $[ 0,4 ]$. II. There exists $a \in [ 0,4 ]$ such that $f ( a ) = 2$. III. $\lim _ { x \rightarrow 1 ^ { - } } ( f \circ f ) ( x ) = 1$.
Which of the following statements are true?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

$$f ( x ) = \left| \frac { 2 x - 1 } { x - 1 } \right|$$
The graph of the function intersects its horizontal asymptote at the point (a, b).
Accordingly, what is the sum $a + b$?
A) $\frac { 5 } { 2 }$
B) $\frac { 7 } { 2 }$
C) $\frac { 8 } { 3 }$
D) $\frac { 9 } { 4 }$

A function $f$ on the set of real numbers is defined as $$f ( x ) = \frac { | x | } { 1 + | x | }$$ Accordingly, which of the following is the image set of the interval $[ - 2,1 )$ under the function $\mathbf{f}$?\ A) $[ 0,1 ]$\ B) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right]$\ C) $\left[ \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$\ D) $\left[ 0 , \frac { 1 } { 3 } \right]$\ E) $\left[ 0 , \frac { 2 } { 3 } \right]$

In the rectangular coordinate plane, the graphs of functions $f$, $g$, and $h$ are given in the figure.
Accordingly, for a real number $a$ satisfying the condition $0 < a < 2$
I. When $f(a) < g(a)$, then $g(a) < h(a)$ holds. II. When $g(a) < h(a)$, then $h(a) < f(a)$ holds. III. When $h(a) < f(a)$, then $f(a) < g(a)$ holds.
Which of the following statements are true?
A) Only I
B) Only II
C) Only III
D) I and II
E) I and III

In the rectangular coordinate plane, the graph of a function f defined on the interval $[ 0,2 ]$ is given below.
Accordingly, I. $( f \circ f ) ( x ) = 2$ II. $( f \circ f ) ( x ) = 1$ III. $( f \circ f ) ( x ) = 0$ Which of these equalities are satisfied for exactly two different values of x?
A) Only I
B) Only II
C) Only III
D) I and II
E) II and III

Let $0 < x _ { 1 } < x _ { 2 }$. A function f defined on the set of real numbers as
$$f ( x ) = \left( x - x _ { 1 } \right) \left( x - x _ { 2 } \right)$$
The parabola represented by this function intersects the axes at different points A and B in the rectangular coordinate plane as shown in the figure.
The distances from points A and B to the origin are equal, and this parabola takes its minimum value when $x = \frac { 3 } { 5 }$. Accordingly, what is the ratio $\frac { \mathbf { x } _ { \mathbf { 2 } } } { \mathbf { x } _ { \mathbf { 1 } } }$?
A) 2
B) 3
C) 4
D) 5
E) 6

In the rectangular coordinate plane, parts of the graphs of functions $f$ and $g$ defined on the closed interval $[0, 7]$ are given in the figure.
On the closed interval $[0, 7]$:

• For 4 different integers $a$, $f(a) < g(a)$,
• For 3 different integers $b$, $f(b) > g(b)$

It is known that. Accordingly, which of the following could be the missing parts of the graphs of functions $f$ and $g$?
A) [Graph A]
B) [Graph B]
C) [Graph C]

In the rectangular coordinate plane, the graph of a function $f$ defined on the closed interval $[-5,5]$ is given in the figure.
For distinct numbers $a, b, c$ and $d$ in the domain of this function
$$\begin{aligned} & f(a) = f(b) = 1 \\ & f(c) = f(d) = 3 \end{aligned}$$
the equalities are satisfied. Accordingly, regarding the ordering of $a, b, c$ and $d$ numbers I. $a < b < c < d$ II. $c < a < b < d$ III. $c < d < a < b$ Which of the following inequalities can be true?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

Where $a, b$ and $c$ are real numbers,
$$y = ax^2 + bx + c$$
the parabola intersects the line $y = 1$ at points B and C, and intersects the line $y = 6$ at only point A. The locations of points A, B and C in the rectangular coordinate plane are shown in the figure below.
Accordingly, what are the signs of the numbers $a$, $b$ and $c$ respectively?
A) +, -, -
B) +, +, -
C) -, +, +
D) -, -, +
E) -, -, -

For real numbers $\mathrm{a}$, $\mathrm{b}$ and $\mathrm{c}$ with $\mathrm{a} \cdot \mathrm{b} \cdot \mathrm{c} > 0$, a function $f$ is defined on the set of real numbers as
$$f(x) = ax^{2} + bx + c$$
Accordingly, the graph of function $f$ can be which of the graphs shown (I, II, III)?
A) Only I B) Only II C) I and III D) II and III E) I, II and III

In the rectangular coordinate plane, the graphs of functions $f$ and $g$ defined on the closed interval $[0,1]$ and the line $y = x$ are given below.
For real numbers $a, b$ and $c$ in the open interval $(0,1)$
$$\begin{aligned} & a < f(a) < g(a) \\ & g(b) < b < f(b) \\ & c < g(c) < f(c) \end{aligned}$$
If these inequalities are satisfied, which of the following orderings is correct?
A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $c < a < b$ E) $c < b < a$

In the rectangular coordinate plane, the graphs of functions $f$, $g$ and $h$ are given in the figure.
For functions $f$, $g$ and $h$
$$(f - g)(1) \cdot (f - h)(1) < 0$$
$$(g - h)(2) \cdot (g - f)(2) > 0$$
Given that the inequalities are satisfied, which of the following orderings is correct?
A) $f(3) < g(3) < h(3)$
B) $f(3) < h(3) < g(3)$
C) $g(3) < f(3) < h(3)$
D) $h(3) < f(3) < g(3)$
E) $h(3) < g(3) < f(3)$

The graph of a function $f$ in the rectangular coordinate plane is given below.
A function $g$ defined on the set of real numbers has a limit at all points where it is defined, and $\lim_{x \rightarrow 3} g(x) = 14$ is calculated.
If the function $f \cdot g$ is continuous on the set of real numbers, what is the value of $g(3)$?
A) 4 B) 6 C) 8 D) 10

Let $a$ and $b$ be real numbers. The function $f$ defined on the set of real numbers as
$$f(x) = \begin{cases} x^{2} - ax + 6 & , x \leq a \\ 2x + a & , a < x \leq b \\ 11 - 2x + b & , x > b \end{cases}$$
is continuous on its domain.
Accordingly, what is the product $a \cdot b$?
A) 4 B) 6 C) 8 D) 10 E) 12