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) $\frac{17}{2}$