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. For each of the following functions: 1. State each relevant transformation, amplitude, and period. 2. Graph one complete period by hand. Be sure to show in the graph the endpoints of one period.
    1. $f(x) = 3\cos\left(2x - \dfrac{\pi}{3}\right)$
    2. $h(x) = \sin(\pi + 2x)$
    3. $y = 5\sin\left(\pi x + \dfrac{\pi^2}{3}\right)$
    4. $y = -5 + 4\cos(\pi - x)$
  2. For each of the following functions: 1. State each relevant transformation. 2. Graph one complete period by hand. Be sure to show in the graph where one period beings and ends.
    1. $y = 5\sec x$
    2. $f(x) = 2\csc \left(\pi x - \dfrac{\pi^2}{2}\right)$
    3. $y = 2\cot\left(x - \dfrac{\pi}{3}\right)$
    4. $y = \frac{1}{2}\tan\left(\pi x\right)$
    5. $y = 4 \tan (3x - 2\pi)$
  3. Suppose \[f(x) = \sin (x) \qquad g(x) = \cos x \qquad h(x) = \tan x\] Find the formula for the following functions:
    1. $f\circ g$
    2. $g\circ f \circ h$
  4. 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. $F(x) = \sec(\tan(x))$
  5. Given a graph of a function, how do you tell if the function is one-to-one?
  6. Because $f(x) = \sin(x)$ is not one to one, what is the interval of values we restrict the domain of $f(x)$ to in order for $f(x)$ to have an inverse?
  7. Inverse function property in 2.8 tells us that if $f(x) = \sin(x)$ and $g(x) = \sin^{-1}(x)$, then $(f\circ g)(x) = x$ is valid. For what values of $x$ is $(f \circ g)(x) = x$ true? Moreover, explain the exact reason why these specific values of $x$ are valid.
  8. Inverse function property in 2.8 tells us that $\cos^{-1}(\cos(x)) = x$. For what values of $x$ is this true? Moreover, explain the exact reason why these specific values of $x$ are valid.
  9. Out of the following, which ones are true? Justify your answer with the domain of the proper function.
    1. $\sin^{-1}\left(\sin\left(\dfrac{\pi}{3}\right)\right) = \dfrac{\pi}{3}$
    2. $\sin^{-1}\left(\sin\left(-\dfrac{\pi}{4}\right)\right) = -\dfrac{\pi}{4}$
    3. $\sin^{-1}\left(\sin\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
    4. $\cos^{-1}\left(\cos\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
    5. $\cos\left(\cos^{-1}\left(\dfrac{1}{2}\right)\right) = \dfrac{1}{2}$
    6. $\tan^{-1}\left(\tan\left(-\dfrac{2\pi}{3}\right)\right) = -\dfrac{2\pi}{3}$
    7. $\tan\left(\tan^{-1}100000000000\right) = 100000000000$
  10. Evaluate the following:
    1. $\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right)$
    2. $\sin^{-1}\left(\dfrac{-1}{2}\right)$
    3. $\cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)$
    4. $\cos^{-1}(2)$
    5. $\tan^{-1}(0)$
    6. $\tan^{-1}(\sqrt{3})$
    7. $\tan\left(\sin^{-1}\frac{\sqrt{2}}{2}\right)$
    8. $\tan^{-1}(1)$
    9. $\sin^{-1}\left(\sin\left(\dfrac{7\pi}{6}\right)\right)$
    10. $\sin^{-1}\left(\dfrac{4}{\pi}\cdot \tan^{-1}\left(1\right)\right)$