Show each step of your work and fully simplify each expression.
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:
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.
$f(x) = 3\cos\left(2x - \dfrac{\pi}{3}\right)$
$h(x) = \sin(\pi + 2x)$
$y = 5\sin\left(\pi x + \dfrac{\pi^2}{3}\right)$
$y = -5 + 4\cos(\pi - x)$
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.
$y = 5\sec x$
$f(x) = 2\csc \left(\pi x - \dfrac{\pi^2}{2}\right)$
$y = 2\cot\left(x - \dfrac{\pi}{3}\right)$
$y = \frac{1}{2}\tan\left(\pi x\right)$
$y = 4 \tan (3x - 2\pi)$
Suppose \[f(x) = \sin (x) \qquad g(x) = \cos x \qquad h(x) = \tan x\]
Find the formula for the following functions:
$f\circ g$
$g\circ f \circ h$
You are given a function $F(x)$. Decompose $F(x)$ into two functions $f, g$ where $F = f \circ g$.
$F(x) = \sin(\cos x)$
$F(x) = \sin^2(x)$
$F(x) = \sin(x^2)$
$F(x) = (x^3 - x^2 - 1)^{2/3}$
$F(x) = (x^2 - x)^2$
$F(x) = \sqrt[5]{(x + 1)^3}$
$F(x) = \sec(\tan(x))$
Given a graph of a function, how do you tell if the function is one-to-one?
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?
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.
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.
Out of the following, which ones are true? Justify your answer with the domain of the proper function.