Let $a$ and $b$ be non-zero integers. A function $f$ is defined on the set of real numbers as
$$f ( x ) = a x + b$$
$$( f \circ f ) ( x ) = f ( x + 2 ) + f ( x )$$
According to this, what is the value of $f(3)$?
A) 7
B) 8
C) 9
D) 10
E) 11

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]

For functions $f$ and $g$ defined on the set of real numbers $$\begin{aligned} & (f \circ g)(x) = x^2 + 3x + 1 \\ & (g \circ f)(x) = x^2 - x + 1 \end{aligned}$$ the equalities are satisfied. Given that $f(2) = 1$, what is the value of $f(3)$?
A) 5
B) 6
C) 7
D) 8
E) 9

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 $a$ and $b$ be real numbers. A function $f$ that is continuous on the set of real numbers is defined as
$$f ( x ) = \begin{cases} x ^ { 2 } - 4 & , x \leq a \\ 5 x - 8 & , a < x \leq b \\ 7 & , x > b \end{cases}$$
Accordingly, what is the sum $a + b$?
A) 4
B) 5
C) 6
D) 7
E) 8

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

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

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

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. 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) $\frac{17}{2}$

Let $a$ be a non-zero real number, and $b$ and $c$ be real numbers. For the function $f(x) = ax + b$ defined on the set of real numbers and its inverse function $f^{-1}$,
$$\begin{aligned} & \lim_{x \rightarrow b} \frac{f(x)}{f^{-1}(x)} = c \\ & f(1) = 3 \end{aligned}$$
are given. Accordingly, what is the sum of the different values that $c$ can take?
A) 6 B) 7 C) 10 D) 11 E) 14