Homework 1

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 a term? How about a factor?
  2. When using the word "term" or "factor", what do you need to specify alongside the word?
  3. In the expression \[-42(x^2 + 3)(x+3)^2 + 5(x+4)^4(x-3)^2\] in what context is the expression $(x^2 + 3)$ not considered a factor, even though it is visually next to a multiplication?
  4. In fraction property #5, which says \[\dfrac{ac}{bc} = \dfrac{a}{b}\] what does $c$ need to be in order to be cancelled out?
  5. Can I cross out the $x^2$ in \[\dfrac{x^2 + 1}{x^2 + 2}\] to get $\dfrac{1}{2}$? Give the reason why or why not.
  6. Can I cross out the $x - 1$ in \[\dfrac{(x-1)(x+2)}{(x-1)(x+3)}\] to get $\dfrac{x+2}{x+3}$? Give the reason why or why not.
  7. Can I cross out the $x - 1$ and $x + 3$ in \[\dfrac{(x-1)(x+2) + 4(x+3)}{(x-1)(x+3)}\] to get $\dfrac{x+2 + 4}{1}$? Give the reason why or why not.
  8. Your friend tries to simplify $(x + 3)(x + 2)$ by using the Distributive law. They treat $(x + 3)$ as a factor and distribute to $x$ and $2$: \[(x + 3)(x + 2) = x + 3 \cdot x + x + 3 \cdot 2\] What did they do incorrectly?
  9. In the global context, is the expression \[-x(x-2)(x+3) + 5y\] comprised of terms or factors?
  10. Write down one fractional expression which satisfies the following:
    • Global context of numerator comprises of three terms
    • Global context of denominator comprises of two terms
    • Each term in the numerator contains two factors
    • Each term in the denominator contains three factors
  11. For each of the following sets, draw their real line representation.
    See Lecture Note I for examples.
    1. $(-\infty, 1) \cup (1, 2) \cup (2, \infty)$
    2. $(-\infty, -6]\cup (2, 10)$
    3. $(-10, -4]\cup (4, \infty)$
  12. A student tries to simplify \[\dfrac{x+3}{x + 2} \cdot 4 = \dfrac{x+3\cdot 4}{x+2} = \dfrac{x+12}{x+2}\] Why is this incorrect?
  13. A student tries to simplify $x^2 + x^3$ by applying exponent laws. They write \[x^2 + x^3 = x^{2+3} = x^5\] Why is this incorrect?
  14. A student tries to simplify $x^2 \cdot x^3$ by applying exponent laws. They write \[x^2 \cdot x^3 = x^{2\cdot3} = x^6\] Why is this incorrect?
  15. A student tries to simplify $(a + b)^2$ by applying exponent laws. They write \[(a + b)^2 = a^2 + b^2\] Why is this incorrect?
  16. A student tries to simplify $(2x + \sqrt{x})^2$ by applying exponent laws. They write \[(2x + \sqrt{x})^2 = 2x^2 + x\] State the two errors they made and why they are incorrect.
  17. True or false: Like terms are expressions which share the same factors, except possibly for the coefficient (the number).
  18. State whether each pair of expressions are like terms or not.
    1. $3x^2$ and $4y$
    2. $3x^2$ and $4x$
    3. $x^3y$ and $4x^3y$
    4. $5(x+1)(x+2)$ and $-(x+1)(x+2)$
    5. $-100(3x-2)(4x^2+3)^2$ and $4(4x^2+3)^2(3x-2)$
  19. Expand and simplify each expression by using the distributive law (or FOIL) and combining like terms.
    See Lecture Note III for expanding examples.
    1. $(2x^2 + 3x) + (3x^3 + 2x)$
    2. $(x+1)(x-2)$
    3. $(x^2 + 2x + 1)(x-2)$
    4. $(1 - x)^2$
    5. $(x^6 - x^5) -2(x^4 - x^3) - x(x^2 - x)$
    6. $3(x+h)^2 - 1 - (3x^2 - 1)$
  20. Factor the following expressions.
    See Lecture Note III for factoring examples.
    1. $-2x^3 - x^2$
    2. $(x+3)^2(x-2) + (x+3)(x-2)^2$
    3. $x^2 - 1$
    4. $x^2 + 5x + 6$
    5. $x^2 + 13x + 12$
    6. $2x^2 + 7x + 3$
    7. $2x^2(x-1) + 7x(x-1) + 3(x-1)$
    8. $4a^2 - 9b^2$
    9. $(x^2 + 1)^2 - 7(x^2 + 1) + 10$
    10. $x^3 + 4x^2 + x + 4$
  21. Rationalize (examples in Lecture Note IV) the numerator for the expression \[\dfrac{\sqrt{x + h} - \sqrt{x}}{h}\]
  22. Rationalize the denominator for the expression \[\dfrac{16x - x^2}{4 - \sqrt{x}}\]
  23. Rationalize the denominator for the expression \[\dfrac{x^2}{\sqrt{x^2 + 9} - 3}\]
  24. Suppose your professor tells you to simplify the expression \[\dfrac{\dfrac{2}{x-2} - \dfrac{3}{x - 3}}{1 + \dfrac{1}{x + 4}}\]
    1. What type of expression is this called?
    2. How should you think about approaching this type of expression?
  25. Perform the indicated operation and fully simplify (meaning write as one fraction only, get rid of negative exponents, remove all compound fractions).
    See Lecture Note IV for fraction manipulation examples and compound fractions.
    1. $\dfrac{3}{10} + \dfrac{4}{15}$
    2. $\dfrac{\sqrt{x^5}}{x^{-1/5}}$
    3. $\dfrac{1}{x + 1} - \dfrac{1}{x - 1}$
    4. $\dfrac{1 + \dfrac{1}{x}}{\dfrac{1}{x} - 2}$
    5. $\dfrac{x}{x + 3} - \dfrac{18}{x^2 - 9}$
    6. $\dfrac{1}{x-1} + \dfrac{1}{x} - \dfrac{1}{x-1}$
    7. $\dfrac{4(x+3)(x-1)}{2(x-1)}$
    8. $\dfrac{x^2 + 2x + 1}{x^2 - 1}$
    9. $\dfrac{x^2h + 2xh + h}{h}$
    10. $2 + \dfrac{1}{x + 3}$
    11. $\dfrac{(x+h)^2 - x^2}{h}$
    12. $\dfrac{\dfrac{1}{x + h} - \dfrac{1}{x}}{h}$
  26. Draw a coordinate plane and graph the points $(1,2), (2, -1), (-3, -4)$ and $(4, -3)$.
  27. 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?
  28. Describe two examples of functions which you encounter in your daily life.
  29. Let $f(x) = x^2 - x + 1$. Evaluate and simplify the following:
    1. $f(1)$
    2. $f(-a)$
    3. $f(x + h)$
    4. $f(x + h) - f(x)$
    5. $f(-x^2)$
  30. Let $f(x) = \dfrac{1}{x + 1}$ Evaluate and simplify the following:
    1. $f(-1)$
    2. $f(x + h)$
    3. $f(x + h) - f(x)$
  31. Consider the expression $f(x)g(x)$.
    1. Suppose $g(x) = x^2 + x + 2$. Substitute $g(x)$ into the above expression correctly.
    2. Now suppose \[f(x) = -x+2 \qquad g(x) = x^2 - 3\] Substitute $f(x)$ and $g(x)$ into the above expression correctly and fully simplify.
    3. When substituting two or more terms into a (local/global) factor, what do you need to not forget?
  32. 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. $f(3)$
  33. Draw a rough sketch of the previous graph by hand. You may verify your graph in Desmos.
    Hint: To graph a piecewise function in Desmos, you need to specify a domain restriction.
    You can graph $3x + 2, x \geq 2$ by typing the symbols \[3x + 2 \{x \geq 2\}\] into Desmos, where the $\geq$ sign is created by first typing $>$ then $=$.
  34. Graph the function by hand: \[f(x) = \begin{cases} -x^2 & x > 0 \\ -x - 1 & x \leq 0 \end{cases}\]
  35. Graph the function by hand: \[g(x) = \begin{cases} 2x & x > 0 \\ x & x < 0 \\ 3 & x = 0\end{cases}\]
  36. Draw one curve in the plane that isn't the graph of a function.
  37. Suppose $f(x)$ and $g(x)$ are two different functions. In the expression $f(x)g(x)$, is $f(x)$ a term or a factor in the global context?
  38. Suppose $f(x) = mx + b$ where $m$ and $b$ are real numbers. Draw a rough sketch of $f(x)$ for each of the following cases:
    1. $m = 0$
    2. $m > 0$
    3. $m < 0$
  39. Suppose you have a function $f(x)$. What is the tangent problem?
  40. Suppose $y = mx + b$ is the equation of the tangent line of the function $f(x)$. Numerically speaking, what is the derivative?
  41. How are the tangent problem and the instantaneous velocity problem related?
  42. In the tangent problem, when the secant line's point moves closer and closer to the point of tangency, what happens to the secant line?

    43. The following problems will appear in a future homework assignment, even though they are precalculus. They are not due for Homework 1.

    I didn't want the homework to be too long.

    Try these out for practice if you have the time.

  43. Redefine the following rational functions with their domain exclusions (holes):
    1. $f(x) = \dfrac{x^2 - 1}{x-1}$
    2. $f(x) = \dfrac{2x^2 + 3x - 2}{x + 2}$
    3. $f(x) = \dfrac{x^3 - 3x^2 - 4x}{x}$
  44. You are given a function $F(x)$. Decompose $F(x)$ into two functions $f, g$ where $F = f \circ g$.
    1. $F(x) = \sin(\cos x)$
    2. $F(x) = \sin^2(x)$
    3. $F(x) = \sin(x^2)$
    4. $F(x) = (x^3 - x^2 - 1)^{2/3}$
    5. $F(x) = (x^2 - x)^2$
    6. $F(x) = \sqrt[5]{(x + 1)^3}$
    7. $F(x) = \sec(\tan(x))$
  45. Find the domain of the following functions:
    1. $f(x) = \dfrac{1}{x^2 - 1}$
    2. $f(x) = \sqrt{x} + \dfrac{1}{x}$
    3. $f(x) = \sin(x)$
    4. $f(x) = \dfrac{1}{\sqrt{x}}$
      Hint: Both problems are present.
  46. Read these two lecture notes on trigonometric functions: 5.1: Unit Circle and 5.2: Trigonometric Functions. This is mandatory reading if you do not remember trigonometric functions.
    Draw a checkmark to get credit for this problem.
  47. Using the unit circle, explain why $\tan\left(\frac{\pi}{2}\right)$ is undefined.
  48. Find the six trigonometric functions of $t = \frac{2\pi}{3}$.
  49. Find the six trigonometric functions of $t = -\frac{5\pi}{6}$.
  50. Suppose \[f(x) = \sin(x) + \cos(x) \qquad g(x) = \tan(x)\] Evaluate and simplify the following expressions without a calculator/the internet:
    1. $f(0)$
    2. $g(0)$
    3. $f\left(\frac{\pi}{2}\right) + g(\pi)$
    4. $f(x)\cdot g(x)$
    5. $f(x) - g(x)$
  51. Suppose \[f(x) = \sin(x) \qquad g(x) = x^2 - x \qquad h(x) = f(x)g(x)\] Evaluate the following:
    1. $f\circ g$
    2. $g \circ f$
    3. $h \circ f$