Homework 1

Directions: Show each step of your work. Turn in your answers in class on a physical piece of paper. Staple multiple sheets together. Feel free to use Desmos for graphing.

Answer the following:

  1. Suppose $A = (-\infty, -2), B = [-5, 1), C = (-6, 4]$. Find the following:
    1. $A\cup B$
    2. $A \cap C$
    3. $A \cup B \cup C$
    4. $(A\cap B) \cup C$
  2. Suppose there is a variable $x$ which is associated with a value $f(x)$. I find that two different inputs give the same evaluation. In particular, I find $x = -2$ and $x = 2$ have $f(-2) = f(2)$. Is $f(x)$ a function?
  3. 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?
  4. Suppose you have an expression $xy(x^2y - xy)$. You expand it by doing $xy \cdot x^2 y - xy \cdot xy = x^3y^2 - x^2y^2$. What property did you use?
  5. Suppose there is an expression \[f(x) = \dfrac{(a + h) + x}{(a+h)^3 - 5}\] Am I allowed to cancel out the $a+h$'s to get \[\dfrac{x}{(a+h)^2 - 5}\] If I am not, when am I allowed to cancel in this fashion?
  6. A student tries to simplify $x^2 + x^3$ by applying exponent laws. They write \[x^2 + x^3 = x^{2+3} = x^5\] State the reason why this is incorrect.
  7. A student tries to simplify $x^2 \cdot x^3$ by applying exponent laws. They write \[x^2 \cdot x^3 = x^{2\cdot3} = x^6\] State the reason why this is incorrect and write the correct answer.
  8. A student tries to simplify $(a + b)^2$ by applying exponent laws. They write \[(a + b)^2 = a^2 + b^2\] State the reason why this is incorrect and write the correct answer.
  9. When finding the domain of a function, what are the two types of inputs we need to exclude?
  10. Suppose I have a function $f(x) = \dfrac{x}{(x+1)^2}$.
    1. Find $f(1), f(-a), f(a+h), f(x+1)$ and fully expand each expression.
    2. When finding the domain, what value do we need to exclude? Why do we need to exclude it?
  11. Suppose a function takes an input $x$ and sends this to $\frac{1}{x + 1}$. What is the domain of $f(x)$?
  12. Draw a coordinate plane and graph the functions $f(x) = x^2, g(x) = x^4$ and $h(x) = x^6$. What is similar between the graphs?
  13. Rewrite the expression $\dfrac{1}{x^2y^3}$ in terms of negative exponents.
  14. Graph the function \[f(x) = \begin{cases}-x^2 & x \leq 3 \\ -x + 1 & x > 3\end{cases}\]
  15. In San Luis Obispo, the minimum sales tax rate $T$ is $8.75\%$ of the value of goods $x$.
    1. Express $T$ as a function of $x$.
    2. Find $T(100)$ and $T(1000)$.
    3. In English, what is the meaning of $T(1000) - T(100)$?
  16. According to the IRS, tax refund fraud from identity theft has been increasing since 2010. The IRS estimates a function to measure the number (in millions of incidents) starting from 2010 $(t = 0)$ through 2013 $(t = 3)$ as \[f(t) = 0.01t^2 + 0.5 \qquad 0 \leq t \leq 3\] Here, $f(t)$ means the total number of incidents in year $t$ only.
    1. How many incidents occured in 2010? How about 2013?
    2. How many more incidents occured in 2013 compared with 2010?
  17. Find two sets $A$ and $B$ such that $A \cap B = [-10, 5)$.
  18. Suppose \[f(x) = x + 1 \qquad g(x) = \dfrac{x^2 + x}{x}\] Find the domains of $f(x)$ and $g(x)$.
  19. Suppose U-Haul rents its 10-ft box truck at $\$45$/day and $\$0.5$/mile. Enterprise rents its 10-ft box truck at $\$60$/day and $\$0.4$/mile.
    1. If $x$ is the number of miles driven, find a function $f(x)$ that represents the daily cost of leasing from U-Haul.
    2. If $x$ is the number of miles driven, find a function $g(x)$ that represents the daily cost of leasing from Enterprise.
    3. Enterprise says our rental is cheaper than U-Haul after 40 miles of driving. Is this claim true or false? Verify with calculations.