Piecewise-Defined Functions

1. Consider the function $f$ defined piecewise as: $$f(x) = \begin{cases} -x &\text{ if } x \leq -8 \text{ or } x \gt 4 \\ \frac{1}{2}(x+7)^{1.61} &\text{ if } -7 \leq x \leq -1 \\ (x-3)^2-5 &\text{ if } -1 \lt x \leq 4 \end{cases}$$
   1. What are the values of the following?

      $f(-8)$

      $f(4)$

      $f(0)$

      $f(-1)$

      $f(1)$

      $f(-7)$

      $f(5)$

      $f(-3)$

   2. Based on its definition what can we infer the domain of $f$ to be?
   3. Graph the function $y = f(x)$ on the axes below, and after you do, conclude what the range of $f$ must be.

      

2. What value(s) might $x$ be if $\left|x^2-3x\right| = 2(x-1)?$
3. In 2024, the city of Grand Junction charges residents for city water use on a tiered system: the more water you use, the higher the rate you’re charged per gallon. Their rates can be distilled into the following piecewise-defined function, where a resident is charged $C(w)$ dollars for using $w$ thousand gallons of water that month. $$C(w) = \begin{cases} 14.81 + 3.17w &\text{ if } 0 \lt w \leq 2 \\21.15 + 4.01(w-2) &\text{ if } 2 \lt w \leq 10 \\53.23 + 4.75(w-10) &\text{ if } 10 \lt w \leq 20 \\100.73 + 5.84(w-20) &\text{ if } 20 \lt w \end{cases}$$
   1. Graph the function $y = C(w).$ Be sure to have some forethought on the scale of your $x$- and $y$-axis before you begin.

      

   2. How much will GJ charge a resident that uses $1600$ gallons of water in a month? What about a resident that uses $7200$ gallons?
   3. Suppose a GJ resident tells you they had to pay $88 on their water bill. How many gallons of water did they use that month?

Challenge

Similar to the tiers at which Grand Junction residents are charged per water usage, the US government sets annual tax rates according to a tiered structure. The tiers in this case are called tax brackets.

Look up the standard deduction (and understand what it is) and the tax rates per tax bracket for this year, and come up with a function $f,$ which must be defined piecewise, that will describe exactly how much a single person with $x$ dollars of income for the year who takes the standard deduction will have to pay in taxes.