Homework 4

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. Isolate $y'$ in the equation \[2y^2y' - 2(xy + y') = 0\]
  2. Solve the following equations. Fully simplify all complex solutions and remember to check solutions.
    1. $\dfrac{1}{x-1} - \dfrac{2}{x^2} = 0$
    2. $\sqrt{8x - 1} = 3$
    3. $\sqrt{2x + 1} + 1 = x$
    4. $\dfrac{2}{\sqrt{4x - 1}} - \dfrac{1}{x} = 0$
    5. $3x^2 + 1 = 0$
    6. $x^2 + 2x + 2 = 0$
  3. Using the discriminant $b^2 - 4ac$, determine whether the following equations have two real solutions, one real solution, or two distinct complex solutions.
    1. $3x^2 - 5x = -2$
    2. $x^2 + 1 = 0$
    3. $x^2 - 2x - 1 = 0$
    4. $2x^2 - x = x^2 + 1$
  4. Simplify the following:
    1. $\sqrt{-4}$
    2. $(-25)^\frac{1}{2}$
    3. $\sqrt{-2}\cdot\sqrt{6}\cdot\sqrt{-3}$
    4. $\dfrac{\sqrt{-49}}{7}$
    5. $\dfrac{\sqrt{-4}}{3}\cdot \dfrac{\sqrt{-3}}{2}$
  5. Solve the following inequalities:
    1. $2x \geq 10$
    2. $5 - 2x < -15$
    3. $-1 < 2x - 5 < 7$
    4. $2 \leq -x + 5 < 4$
    5. $2(7x - 3) < 12x + 16$
  6. Draw a coordinate plane and graph the points $(1,2), (2, -1), (-3, -4)$ and $(4, -3)$.
  7. Sketch a graph of the following equations by picking points:
    1. $x - y = 1$
    2. $y = \sqrt{x}$
      Hint Pick perfect squares for $x$, i.e. $1, 4, 9, \dots$
    3. $y - x = -2$
    4. $x^2 - y = 1$
    5. $-\lvert x \rvert - 1 + y = 0$
  8. Find the $x-$ and $y-$ intercepts of the following equations.
    1. $y = x + 6$
    2. $y = x^2 - 4$
    3. $y = -\sqrt{16 - x^2}$
  9. Draw two lines in the coordinate plane; one with negative slope and one with positive slope.
  10. Find the equation of a line that passes through $(4, 4)$ and $(0, 2)$. Write your answer in slope-intercept form.
  11. Find the equation of a line that passes through $(-2, 5)$ and $(0, 2)$. Write your answer in slope-intercept form.
  12. Find the equation of a line with slope $-3$ and passes through the point $(1,1)$. Write your answer in slope-intercept form.
  13. Find the equation of a vertical line that passes through $(-2, 4)$.
  14. Find the equation of a horizontal line that passes through $(-3, -3)$.
  15. Suppose I have an expression $f(x)$ and I find two different inputs that give the same evaluation. In particular, I find that $x = 2$ and $x = 3$ gives $f(2) = f(3)$. Is $f$ a function?
  16. Suppose I have an expression $f(x)$ and I find one input which gives two different evaluations. In particular, I find that $x = 2$ spits out $f(2) = 5$ and $f(2) = 3$. Is $f$ a function?
  17. Write down two examples of functions in your daily life.
  18. Let $f(x) = x^2 - x + 1$. Evaluate and simplify the following:
    1. $f(1)$
    2. $f(a)$
    3. $f(-a)$
    4. $f(x + h)$
    5. $\dfrac{f(x + h) - f(x)}{h}$
  19. Let $f(x) = \dfrac{1}{x}$. Evaluate and simplify the following:
    1. $f(-1)$
    2. $f(x + h)$
    3. $\dfrac{f(x + h) - f(x)}{h}$
  20. Let $f(x) = 2x^2 - x$. Evaluate and simplify the following:
    1. $f(0)$
    2. $f(x + h)$
    3. $\dfrac{f(x + h) - f(x)}{h}$
  21. Suppose $f$ is a function. What two types of numbers do you need to remove when finding the domain?
  22. Find the domain of the following functions:
    1. $f(x) = \dfrac{1}{x - 1}$
    2. $f(x) = \dfrac{1}{x^2 + 3x + 2}$
    3. $f(x) = \dfrac{x^2 - 2x + 1}{x - 1}$
    4. $f(x) = \dfrac{x^2}{(x^2-1)(x^2-4)}$
    5. $f(x) = \sqrt{2x - 1}$