Homework 3

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. Draw a curve in the plane that is not a function.
  2. Given the functions $f(x) = \sqrt{x - 2}$ and $g(x) = 5 - x^2$, find the following and fully simplify:
    1. $f(g(0))$
    2. $f\circ g$
    3. $g \circ f$
    4. $f \circ f$
    5. $g \circ g$
  3. Given the following functions $F(x)$, find two functions $f$ and $g$ where $f\circ g = F$. You are not allowed to choose $f(x) = x$ or $g(x) = x$.
    1. $F(x) = (x-3)^{-\frac{1}{2}}$
    2. $F(x) = \sqrt{x - 2} + \dfrac{1}{\sqrt{x - 2}}$
    3. $F(x) = \sqrt{(x^2 + 2x + 3)^3}$
  4. 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.
  5. 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.
  6. Draw a graph of a one-to-one function.
  7. Suppose $f$ is not a one-to-one function. Explain what property is violated when we try to define $f^{-1}$.
  8. Suppose $f(x) = x$ and $g(x) = \dfrac{1}{x}$. Are $f(x)$ and $g(x)$ inverses? Show using the Inverse Function Property.
  9. Suppose $f(x) = x^2$ and $g(x) = \sqrt{x}$. Are $f(x)$ and $g(x)$ inverses? Show using the Inverse Function Property.
  10. Is the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ on the unit circle? Show using calculations.
  11. What is the reference number $\bar{t}$? What is the range of values $\bar{t}$ is allowed to be?
  12. Find the following terminal points $P$ associated with the following $t$ values:
    1. $t = 0$
    2. $t = \frac{3\pi}{2}$
    3. $t = \frac{2\pi}{3}$
    4. $t = \frac{5\pi}{6}$
    5. $t = -\frac{2\pi}{3}$
    6. $t = -\frac{9\pi}{4}$
    7. $t = -\frac{11\pi}{6}$
    8. $t = 1000\pi + \frac{2\pi}{3}$
    9. $t = -83\pi + \frac{\pi}{3}$
    10. $t = \pi + 2\pi - 3\pi + \frac{4\pi}{3}$
  13. Find the six trigonometric functions of $t = \frac{2\pi}{3}$.
  14. Find the six trigonometric functions of $t = -\frac{5\pi}{6}$.
  15. Explain using the terminal point of $t = \frac{\pi}{2}$ why $\tan\left(\frac{\pi}{2}\right)$ does not exist.
  16. What is the domain of $f(t) = \sin(t) \cdot \cos(t)$? Hint: section 2.7.
  17. What is the domain of $f(t) = \sin(t) + \tan(t)$?
  18. A function usually has the notation $y = f(x)$, meaning you plug in $x$ and the result is called $y$.
    When we say for any $t \in \mathbb{R}$ and it's associated terminal point $P(x,y)$, what does $\sin t = x$ mean in English?
  19. Find the following:
    1. $\cos\left(\dfrac{10\pi}{3}\right)$
    2. $\sin\left(\dfrac{4\pi}{3}\right)$
  20. If $\sin t > 0$ and $ \cos t < 0$, what quadrant is the terminal point in?
  21. Graph $f(t) = \cos t$ by hand.