Homework 2
Directions:
 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:
 Find the six trigonometric functions of $t = \frac{2\pi}{3}$.
 Find the six trigonometric functions of $t = \frac{5\pi}{6}$.
 Why is $\tan\left(\frac{\pi}{2}\right)$ undefined?

Suppose \[f(x) = x^2 + 2 \qquad g(x) = \dfrac{x + 1}{x  1}\]
 Find $f\circ g, g\circ f, (f\circ g)(5), (g\circ f)(5)$.
 For the next few parts, find \[\dfrac{f(a + h)  f(a)}{h},\qquad h \neq 0\] and simplify fully:
 $f(x) = x^2$
 $f(x) = \dfrac{1}{x}$
 $f(x) = x^3  1$
 For the next few parts, find two functions $f$ and $g$ such that $h = f\circ g$. You are not allowed to choose $f(x) = x$ or $g(x) = x$.
 $h(x) = (x^3 + x + 1)^3$
 $h(x) = \dfrac{1}{\sqrt{x^2  2}}$
 $h(x) = \dfrac{1}{\sqrt{x + 1}} + \sqrt{x + 1}$
 Use the law of exponents to fully simplify \[\dfrac{\sqrt{a\sqrt{b}}}{\sqrt[3]{ab}}\]
 Solve each equation for $x$:
 $\ln(x^2  1) = 3$
 $e^{2x}5e^x + 4 = 0$
 $2^{x5}=3$

If $f$ is a onetoone function and $f(6) = 17$, what is $f^{1}(17)$?
 Show why $f^{1}$ is not the same function as $\dfrac{1}{f}$ with an example. Hint: The function $f(x) = x^n$
 If $f(x) = x^2, x > 0$, find $f(f^{1}(2))$.
 Combine into one logarithm:
 $2 \ln 5 + 3 \ln 2$
 $\ln(a + b)  \ln(ab)  2\ln c$
 List the first five terms of these sequences. Make sure you have a list of numbers.
 $a_n = \dfrac{2n}{n^2 + 1}$
 $a_n = \cos\dfrac{n\pi}{2}$
 Find a formula for $a_n$ for the following sequences:
 $1, 0, 1,0, 1, 0, 1, 0$
 $\dfrac{1}{2}, \dfrac{4}{3}, \dfrac{9}{4},  \dfrac{16}{5}, \dfrac{25}{6}, \dots$