Question 164
Uma função $f$ é definida por $f(x) = 2^x$. O valor de $f(3) - f(1)$ é
(A) 2 (B) 4 (C) 6 (D) 8 (E) 16

Uma função exponencial é definida por $f(x) = 2^x$. O valor de $f(3) - f(1)$ é
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

When the two roots of the equation $4 ^ { x } - 7 \cdot 2 ^ { x } + 12 = 0$ are $\alpha , \beta$, find the value of $2 ^ { 2 \alpha } + 2 ^ { 2 \beta }$. [3 points]

Find the sum of all natural numbers $x$ satisfying the inequality $\left( \frac { 1 } { 2 } \right) ^ { x - 5 } \geq 4$. [3 points]

How many natural numbers $x$ satisfy the inequality $\left( \frac { 1 } { 9 } \right) ^ { x } < 3 ^ { 21 - 4 x }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

How many natural numbers $x$ satisfy the inequality $\left( \frac { 1 } { 9 } \right) ^ { x } < 3 ^ { 21 - 4 x }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For a natural number $n$, define the function $f ( x )$ as $$f ( x ) = \begin{cases} \left| 3 ^ { x + 2 } - n \right| & ( x < 0 ) \\ \left| \log _ { 2 } ( x + 4 ) - n \right| & ( x \geq 0 ) \end{cases}$$ Let $g ( t )$ be the number of distinct real roots of the equation $f ( x ) = t$ for a real number $t$. Find the sum of all natural numbers $n$ such that the maximum value of the function $g ( t )$ is 4. [4 points]

14. Let the function $f ( x ) = \left\{ \begin{array} { c c } 2 ^ { x } - a , & x < 1 , \\ 4 ( x - a ) ( x - 2 a ) , & x \geqslant 1 \text { .} \end{array} \right.$
(1) If $a = 1$, then the minimum value of $f ( x )$ is $\_\_\_\_$;
(2) If $f ( x )$ has exactly 2 zeros, then the range of the real number $a$ is $\_\_\_\_$.
III. Answer Questions (6 questions in total, 80 points. Solutions should include written explanations, calculation steps, or proof processes)

7. The solution set of the inequality $2 ^ { x ^ { 2 } - x } < 4$ is $\_\_\_\_$ .

Given sets $A = \{ x \mid x < 1 \} , B = \left\{ x \mid 3 ^ { x } < 1 \right\}$, then
A. $A \cap B = \{ x \mid x < 0 \}$
B. $A \cup B = \mathbf { R }$
C. $A \cup B = \{ x \mid x > 1 \}$
D. $A \cap B = \varnothing$

5. The sum of the zeros of the function $f ( x ) = \left\{ \begin{array} { l } 6 ^ { x } - 2 , x > 0 , \\ x + \log _ { 6 } 12 , x \leq 0 \end{array} \right.$ is
A. $-1$ B. $1$ C. $-2$ D. $2$

The set of all real numbers $x$ for which $3^{2^{1-x^2}}$ is an integer has
(A) 3 elements
(B) 15 elements
(C) 24 elements
(D) infinitely many elements

If $f ( x ) = \left( \frac { 3 } { 5 } \right) ^ { x } + \left( \frac { 4 } { 5 } \right) ^ { x } - 1 , x \in R$, then the equation $f ( x ) = 0$ has:
(1) No solution
(2) More than two solutions
(3) One solution
(4) Two solutions

The minimum value of $f ( x ) = a ^ { a ^ { x } } + a ^ { 1 - a ^ { x } }$, where $a , x \in R$ and $a > 0$, is equal to:
(1) $a + 1$
(2) $2 a$
(3) $a + \frac { 1 } { a }$
(4) $2 \sqrt { a }$

The equation $e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0, x \in R$ has:
(1) four solutions two of which are negative
(2) two solutions and both are negative
(3) no solution
(4) two solutions and only one of them is negative

Let $S = \{x : x \in \mathbb{R}$ and $\left(\sqrt{3} + \sqrt{2}\right)^{x^2 - 4} + \left(\sqrt{3} - \sqrt{2}\right)^{x^2 - 4} = 10\}$. Then $n(S)$ is equal to
(1) 2
(2) 4
(3) 6
(4) 0

Let x be a real number such that
$$( \sqrt { 7 } + \sqrt { 3 } ) ^ { x } = 4$$
Given this, which of the following is the expression $( \sqrt { 7 } - \sqrt { 3 } ) ^ { x }$ equal to?
A) $2 ^ { - x }$
B) $2 ^ { - x + 1 }$
C) $4 ^ { x }$
D) $4 ^ { x - 1 }$
E) $4 ^ { x + 1 }$

The operation $\Delta$ is defined on the set of real numbers for all real numbers a and b as
$$a \Delta b = a ^ { 2 } + 2 ^ { b }$$
Given that $2 \Delta ( 1 \Delta x ) = 12$, what is x?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 1 } { 4 }$
D) 1
E) 2

For integers a and b
$$\frac { 6 ^ { a ^ { 2 } + b ^ { 2 } } } { 9 ^ { a b } } = 96$$
Given that, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 1 B) 2 C) 3 D) 4 E) 6

$$\begin{aligned} & 4 ^ { x } + 4 ^ { y } = 10 \\ & 4 ^ { x } - 4 ^ { y } = 8 \end{aligned}$$
Accordingly, what is the value of the expression $\mathbf { 2 } ^ { \mathbf { x } + \mathbf { y } }$?
A) 2 B) 3 C) 4 D) 5 E) 6