Homework 2

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. What is the definition of like terms?
  2. Expand and simplify each expression.
    1. $(2x^2 + 3x) + (3x^3 + 2x)$
    2. $(x+1)(x-2)$
    3. $(x^2 + 2x + 1)(x-2)$
    4. $(1 - x)^2$
    5. $(x^6 - x^5) -2(x^4 - x^3) - x(x^2 - x)$
  3. Factor the following expressions.
    1. $-2x^3 - x^2$
    2. $(x+3)^2(x-2) + (x+3)(x-2)^2$
    3. $x^2 + 5x + 6$
    4. $x^2 + 13x + 12$
    5. $2x^2 + 7x + 3$
    6. $2x^2yz + 7xyz + 3yz$
    7. $3x^4 - 16x^2 + 5$
    8. $(3x+2)^2 + 8(3x+2) + 12$
  4. Isolate $x$ in the following expressions:
    1. $3x + d = q$
    2. $ax + bx = c$
    3. $3x + b = 4x - 3$
  5. A student tries to simplify \[\dfrac{x^{-1} + y^{-1}}{4} = \dfrac{1}{4xy}\] What mistake did the student make?
  6. A student tries to simplify \[\dfrac{x+3}{x + 2} \cdot 4 = \dfrac{x+3\cdot 4}{x+2} = \dfrac{x+12}{x+2}\] What mistake did the student make?
  7. Show why $x = 2$ is a solution to the equation \[\dfrac{1}{x} - \dfrac{1}{x-4} = 1\]
  8. Find all real-valued solutions for the following equations.
    1. $3x + 4 = 7$
    2. $2x + 3 = 7 - 3x$
    3. $x^2 + x - 12 =0$
    4. $2x^2 = 8$
    5. $\dfrac{1}{x} = \dfrac{4}{3x} + 1$
    6. $\dfrac{1}{x-1} + \dfrac{1}{x+2} = \dfrac{5}{4}$
  9. Isolate $x$ in the equation $a(b + cx) + d = e$. Simplify any compound fractions.
  10. A student tries to isolate $x$ in the equation $(a + b)x = c + d$ by dividing by $a + b$, resulting in \[x = \dfrac{c}{a + b} + d\] What mistake did the student make?
  11. 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?
  12. 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?
  13. Write down two examples of functions in your daily life.
  14. Let $f(x) = x^2 - x + 1$. Evaluate the following:
    1. $f(1)$
    2. $f(a)$
    3. $f(-a)$
    4. $f(x + h)$
  15. Let \[f(x) = \begin{cases} 3x + 2 & x \geq 2 \\ x^2 - x & x < 2\end{cases}\] Evaluate the following:
    1. $f(0)$
    2. $f(1)$
    3. $f(2)$
    4. $f(3)$
  16. Suppose $f$ is a function. What two problems do you need to look for when finding the domain?
  17. Find the domain of the following functions:
    1. $f(x) = \dfrac{1}{x - 1} + \dfrac{1}{x - 2}$
    2. $f(x) = \sqrt{x - 1} + \dfrac{1}{x - 3}$
  18. Graph the points $(1, 2), (-1, 4), (-3,-2)$ and $(3, -4)$ in the coordinate plane.
  19. Sketch a graph of the function $f(x) = x^2-1$ drawn to scale on the coordinate plane.
  20. In the coordinate plane, what physical meaning does $f(x)$ take on?
  21. Draw the general shape of the functions $f(x) = x^2, g(x) = x^4, h(x) = x^6$ in the coordinate plane.
    Do the same for the odd powers $f(x) = x^3, g(x) = x^5, h(x) = x^7$.

    22. The rest of these problems will be on next week's homework. Skip for now.

  22. Draw a curve in the plane that is not a function.
  23. Given the functions $f(x) = \sqrt{x - 2}$ and $g(x) = 5 - x^2$, find the following:
    1. $f(g(0))$
    2. $f\circ g$
    3. $g \circ f$
    4. $f \circ f$
    5. $g \circ g$
  24. Given the function $F(x) = \dfrac{1}{\sqrt{x - 3}}$, write $F(x)$ as a composition of functions $f$ and $g$ (meaning determine $f$ and $g$ such that $F(x) = (f\circ g)(x)$).
  25. Suppose \[f(x) = x \qquad g(x) = \dfrac{1}{x-2} \qquad h(x) = \sqrt{x} \qquad i(x) = \dfrac{1}{x-3}\]
    1. Find $f(x) + g(x) - h(x) - i(x)$ and it's domain.
    2. Find $f(x) \cdot g(x) \cdot h(x) \cdot i(x)$ and it's domain.
  26. Suppose \[f(x) = x^2 \qquad g(x) = \sqrt{x} \qquad h(x) = \dfrac{1}{x - 1} \qquad i(x) = \dfrac{1}{x-2}\]
    1. Find $f(x) + g(x) - h(x) - i(x)$ and it's domain.
    2. Find $\dfrac{f(x) \cdot g(x) \cdot h(x)}{i(x)}$ and it's domain.