Collection of Algebra and Function Questions
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II) choose the best answer
6. Which one of the following is power function? A. $f(x) = 3^x$ B. $f(x) = \log_3 x$ C. $h(x) = 5x^{-2}$ D. $k(x) = \ln x$
7. What is the value of $x$ in floor function $f(x) = \lfloor 2x + 1 \rfloor = 12$.
A. $\frac{11}{2} \le x < 6$ B. $\frac{11}{2} < x < 6$ C. $\frac{11}{2} < x \le 6$ D. $\frac{11}{2} \le x \le 6$
8. Which one of the following is the domain of $f(x) = 5x^4$?
A. $\mathbb{R}$ B. $(-\infty, 0)$ C. $[0, \infty)$ D. $(0, \infty)$
9. Which one of the following is not a function?
A. $R = \{(x, y) : x, y \in \mathbb{R} \text{ and } x^2 - y^2 = 0\}$ C. $R = \{(x, y) : x, y \in \mathbb{R} \text{ and } x^2 + y^2 = 1\}$
B. $R = \{(3, 0), (1, 3), (4, 6), (3, 6)\}$ D. All
10. What is the range of $3sgn(x)$? A. $\{3, 0, -3\}$ B. $\{-1, 0, 1\}$ C. $\mathbb{R}$ D. $[0, \infty)$
11. Which one of the following is the greatest integer of $-\pi$ or $\lfloor -\pi \rfloor$.
A. 4 B. 3 C. -4 D. -5
12. Let $f(x) = 2x^2 + 1$ and $g(x) = \sqrt{x + 1}$, then what is $f \circ g(x)$?
A. $2x + 3$ B. $\sqrt{2x^2 + 2}$ C. $2x - 3$ D. $\sqrt{2x^2 - 2}$
13. Which one of the following are not inverse of each other?
A. $f(x) = \frac{x+2}{x-1}$ and $g(x) = \frac{2x+2}{x-1}$ C. $f(x) = x - 1$ and $g(x) = x + 1$
B. $f(x) = 2x - 3$ and $g(x) = 3 - 2x$ D. $f(x) = 8x$ and $g(x) = \frac{x}{8}$
14. Which one of the following is a polynomial expression?
A. $x^2 + 2x + 1$ B. $2\sqrt{x} + 5x + 1$ C. $3^{-1} + x - 9$ D. All
15. What is the domain of $\frac{2x^2 + 5x + 1}{x^2 + 1}$? A. $\mathbb{R}$ B. $\mathbb{R}^+$ C. $\mathbb{R} \setminus \{-1\}$ D. $\mathbb{Z}$
16. Which one of the following conditions are satisfied for rational inequality
A. $\frac{p(x)}{q(x)} \ge 0$ B. $\frac{p(x)}{q(x)} \le 0$ C. $\frac{p(x)}{q(x)} > 0$ D. All
Animated Video Solution
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Step by Step Written Solution
Let's find the values of x that satisfy this floor function equation. We are given the function f of x equals the floor of two x plus one, and we are told that this equals twelve.
Solving a Floor Function Equation
Recall the definition of the floor function. For any integer n, the floor of some expression u equals n if and only if u is greater than or equal to n, but strictly less than n plus one.
In our problem, the integer n is twelve, and our expression u is two x plus one. So we can rewrite our equation as a double inequality.
Simplifying the right side, we have twelve is less than or equal to two x plus one, which is less than thirteen.
Now, we need to isolate x. We'll start by subtracting one from all parts of the inequality.
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