Algebraic functions and polynomials: introduction

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1. Question 1What is the domain of the function $f(x) = \frac{\sqrt{4 - x}}{x - 1}$?

   The $x$ with $x \leq 4$ and $x \neq 1$.

   The $x$ with $x < 4$ and $x \neq 1$.

   The $x$ with $x \leq 4$.

   The $x$ with $x \geq 4$.

2. Question 2The function $f(x) = (x - 1)^2 + 2$, restricted to $\ico{1}{\infty}$, is bijective onto $\ico{2}{\infty}$. What is its inverse?

   $f^{-1}(y) = 1 + \sqrt{y - 2}$

   $f^{-1}(y) = 1 - \sqrt{y - 2}$

   $f^{-1}(y) = \sqrt{y - 2} - 1$

   $f^{-1}(y) = \frac{1}{(y - 1)^2 + 2}$

3. Question 3For what value of $k$ is the polynomial $x^3 + k x^2 + x - 6$ divisible by $x - 2$?

4. Question 4If $r$ and $s$ are the roots of $2x^2 - 6x + 1$, what is the value of $r^2 + s^2$?

5. Question 5Let $a$ and $b$ be integers. According to the rational root theorem, which of the following numbers cannot be a root of $3x^3 + a x^2 + b x + 2$?

   $\frac{3}{2}$

   $\frac{2}{3}$

   $-\frac{1}{3}$

   $-2$

6. Question 6Which numbers $A$ and $B$ satisfy $\frac{3x + 1}{(x - 1)(x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}$ for every real number other than $1$ and $-2$?

   $A = \frac{4}{3}$, $B = \frac{5}{3}$

   $A = 4$, $B = -5$

   $A = \frac{4}{3}$, $B = -\frac{5}{3}$

   $A = 3$, $B = 1$

7. Question 7What is the solution set of the inequality $\frac{(x - 1)^2 (x + 3)}{x - 2} < 0$?

   The $x$ with $-3 < x < 2$ and $x \neq 1$.

   The $x$ with $-3 < x < 2$.

   The $x$ with $x < -3$ or $1 < x < 2$.

   The $x$ with $x < -3$ or $x > 2$.

8. Question 8What is the oblique asymptote of the graph of $f(x) = \frac{2x^2 + 3x - 1}{x - 1}$?

   $y = 2x + 5$

   $y = 2x + 3$

   $y = 2$

   $x = 1$

9. Question 9What is the asymptote of the graph of $f(x) = \sqrt{x^2 + 6x}$ as $x$ tends to $-\infty$?

   $y = -x - 3$

   $y = x + 3$

   $y = -x + 3$

   $y = -x$

10. Question 10A tank contains $50$ liters of water with $4$ grams of sugar per liter. From a given instant on, $2$ liters per minute of a syrup with $30$ grams of sugar per liter are added to it, without any liquid leaving, and the mixing is immediate. What values does the sugar concentration, in grams per liter, take from that instant on?

    All numbers $y$ with $4 \leq y < 30$.

    All numbers $y$ with $4 \leq y \leq 30$.

    All numbers $y$ with $0 \leq y < 30$.

    All numbers $y$ with $4 \leq y \leq 17$.

11. Question 11From earlier unitsWhat is the solution set of the inequality $\abs{2x - 3} < 5$?

    The $x$ with $-1 < x < 4$.

    The $x$ with $x < 4$.

    The $x$ with $x < -1$ or $x > 4$.

12. Question 12From earlier unitsCompute $27^{-\frac{2}{3}}$.

13. Question 13From earlier unitsWhat is the solution set of the inequality $\frac{x - 1}{3} - \frac{2x + 1}{2} \geq 1$?

    The $x$ with $x \leq -\frac{11}{4}$.

    The $x$ with $x \geq -\frac{11}{4}$.

    The $x$ with $x \leq -\frac{5}{4}$.