Consider the function $g$ defined on $[0; +\infty[$ by $g(t) = \frac{a}{b + \mathrm{e}^{-t}}$ where $a$ and $b$ are two real numbers. We know that $g(0) = 2$ and $\lim_{t \rightarrow +\infty} g(t) = 3$. The values of $a$ and $b$ are:
A. $a = 2$ and $b = 3$
B. $a = 4$ and $b = \frac{4}{3}$
C. $a = 4$ and $b = 1$
D. $a = 6$ and $b = 2$

Two exponential functions $f ( x ) = a ^ { b x - 1 } , g ( x ) = a ^ { 1 - b x }$ satisfy the following conditions.
(a) The graphs of $y = f ( x )$ and $y = g ( x )$ are symmetric with respect to the line $x = 2$.
(b) $f ( 4 ) + g ( 4 ) = \frac { 5 } { 2 }$
What is the value of the sum of the two constants $a + b$? (where $0 < a < 1$) [3 points]
(1) 1
(2) $\frac { 9 } { 8 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 11 } { 8 }$
(5) $\frac { 3 } { 2 }$

When the graph of the exponential function $y = 5 ^ { x - 1 }$ passes through the two points $( a , 5 ) , ( 3 , b )$, find the value of $a + b$. [3 points]

14. If the function $f ( x ) = \left\{ \begin{array} { l } - x + 6 , x \leq 2 , \\ 3 + \log _ { a } x , x > 2 , \end{array} ( a > 0 \right.$ and $a \neq 1 )$ has range $[ 4 , + \infty )$, then the range of the real number $a$ is $\_\_\_\_$.

13. Given that the graph of function $f ( x ) = a x ^ { 3 } - 2 x$ passes through the point $( - 1,4 )$, then $a = $ $\_\_\_\_$ .

Let the function $f ( x ) = \mathrm { e } ^ { x } + a \mathrm { e } ^ { - x }$ ($a$ is a constant). If $f ( x )$ is an odd function, then $a =$ $\_\_\_\_$; if $f ( x )$ is an increasing function on $\mathbb { R }$, then the range of $a$ is $\_\_\_\_$.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \log _ { \sqrt { m } } \{ \sqrt { 2 } ( \sin x - \cos x ) + m - 2 \}$, for some $m$, such that the range of $f$ is $[ 0,2 ]$. Then the value of $m$ is $\_\_\_\_$.
(1) 5
(2) 3
(3) 2
(4) 4

Let $f: R \rightarrow R$ be defined $f(x) = ae^{2x} + be^x + cx$. If $f(0) = -1$, $f'(\log_e 2) = 21$ and $\int_0^{\log 4} (f(x) - cx)\, dx = \frac{39}{2}$, then the value of $|a + b + c|$ equals:
(1) 16
(2) 10
(3) 12
(4) 8