Draw a coordinate plane and graph the points $(1,2), (2, -1), (-3, -4)$ and $(4, -3)$.
Suppose I have an expression $f(x)$ and I find two different inputs that give the same evaluation. In particular, I find that $x = 2$ and $x = 3$ gives $f(2) = f(3)$. Is $f$ a function?
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?
Write down two examples of functions which you encounter in your daily life.
Let $f(x) = x^2 - x + 1$. Evaluate and simplify the following:
$f(1)$
$f(a)$
$f(-a)$
$f(x + h)$
$f(x + h) - f(x)$
$f(-x^2)$
Let $f(x) = \dfrac{1}{x + 1}$ Evaluate and simplify the following:
$f(-1)$
$f(x + h)$
$f(x + h) - f(x)$
Let \[f(x) = \begin{cases} 3x + 2 & x \geq 2 \\ x^2 - x & x < 2\end{cases}\] Evaluate the following:
$f(0)$
$f(1)$
$f(2)$
$f(3)$
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 $=$.
Graph the function by hand: \[f(x) = \begin{cases} -x^2 & x > 0 \\ -x - 1 & x \leq 0 \end{cases}\]
Graph the function by hand: \[g(x) = \begin{cases} \sin(x) & x > 0 \\ x & x < 0 \\ 3 & x = 0\end{cases}\]
Draw one curve in the plane that isn't the graph of a function.
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. Write "Without using the internet, I can evaluate trigonometric functions and know their graph shapes." to get credit for this problem.
For what inputs $x$ does the function $f(x) = \cos(x)$ output $0$? (Hint: 5.1 + 5.2 Lecture notes. In other words, which radian value results in an $x$-coordinate of 0 on the unit circle?).
Find the six trigonometric functions of $t = \frac{2\pi}{3}$.
Find the six trigonometric functions of $t = -\frac{5\pi}{6}$.
Suppose \[f(x) = \sin(x) + \cos(x) \qquad g(x) = \tan(x)\]
Evaluate and simplify the following expressions without a calculator/the internet:
$f(0)$
$g(0)$
$f\left(\frac{\pi}{2}\right) + g(\pi)$
$f(x)\cdot g(x)$
$f(x) - g(x)$
Redefine the following rational functions with their domain exclusions (holes):
$f(x) = \dfrac{x^2 - 1}{x-1}$
$f(x) = \dfrac{2x^2 + 3x - 2}{x + 2}$
$f(x) = \dfrac{x^3 - 3x^2 - 4x}{x}$
Suppose \[f(x) = \sin(x) \qquad g(x) = x^2 - x \qquad h(x) = f(x)g(x)\] Evaluate the following:
$f\circ g$
$g \circ f$
$h \circ f$
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?
Suppose \[f(x) = x^2 - x \qquad g(x) = x^3 - x^2 + 1\]
Evaluate (and always remember to simplify) the following:
$f(x)g(x)$
$f(x)g(x) - [f(x)]^2$
$\dfrac{f(x)[g(x)]^3}{[g(x)]^2}$
Hint: Could you use the previous problem's idea to perhaps simplify your calculation?
You are given a function $F(x)$. Find two functions $f, g$ where $F = f \circ g$.
$F(x) = \sin(\cos x)$
$F(x) = \sin^2(x)$
$F(x) = \sin(x^2)$
$F(x) = (x^3 - x^2 - 1)^{2/3}$
$F(x) = (x^2 - x)^2$
$F(x) = \sqrt[5]{(x + 1)^3}$
$F(x) = \sec(\tan(x))$
Find the domain of the following functions:
$f(x) = \dfrac{1}{x^2 - 1}$
$f(x) = \sqrt{x} + \dfrac{1}{x}$
$f(x) = \tan(x)$
$f(x) = \sin(x)$
$f(x) = \dfrac{1}{\sqrt{x}}$
Hint: Both problems are present.
Suppose $f(x) = mx + b$ where $m$ and $b$ are real numbers. What does the graph of $f(x)$ look like?