More practice for Special Functions – With solutions contributed by students in period 2
For a complete response and full credit, explain your thinking in words; label any figures you
draw; identify and formulas you use; make clear the source of any numbers you use.
1. a) Graph y = 5|x| - 1 without using a graphing calculator. Show reasoning using the algebraic
definition of absolute value. Plotting points alone is not sufficient.
b) Graph y = |2x + 3| without using a graphing calculator. Show reasoning using the algebraic
definition of absolute value. Plotting points alone is not sufficient.
Solution by David & Nico:
2) A silk-screen shop charges the following rates for t-shirt orders.
• An initial charge of $100 to create the silk screen
• $5 per shirt for orders of 50 or fewer shirts
• $4 per shirt for orders of more than 50 shirts
a) How much do you pay if you order 30 shirts? 65 shirts? 80 shirts?
b) Write a piecewise function that gives the total cost given any number of shirts.
c) Draw the graph of the function.
Solution: By Matt & Reed period 2
3) Part of the amount of Social Security Tax (FICA) you pay depends on your annual income. As of 2006,
you pay 6.2% of your income if it is less than $94,200. If you income is at least $94,200 then you pay a
fixed amount of $5840.40
a) How much Social Security Tax do you pay if you make $30,000 a year? $100,000?
b) Write a piecewise function that gives the Social Security Tax given any amount of income.
c) Draw the graph of the function.
Solution: By Matt & Reed period 2
4) You have a summer job that pays time and a half for overtime. If you work more than 40 hours per
week, your overtime wage is 1.5 times your normal wage of $8.00 per hour.
a) What is your income if you work 30 hours? 40 hours? 45 hours?
b) Write a piecewise function that gives your income given any amount of time worked.
c) Draw the graph of the function.
Solution: By Matt & Reed period 2
5) A parking lot charges $2 for the first hour and $1 for each additional hour or part of an hour up to a
maximum charge of $12 per day.
a) How much do you pay if you park for 5 minutes? 4 hours and 2 minutes? 10 hours? 15 hours?
b) Write a piecewise function that gives the total cost of parking given any amount of time during the
day.
c) Draw the graph of the function.
Solution by Lindsay & Kathy:
Comment on Lindsay’s function f(x):
Since part of an hour is counted as a whole hour, the quantity (x-1) in f(x) should be rounded up to the
next integer. This is noted as éêx -1ùú. So the function f(x) should be as follows:
ìï
2 + éêx -1ùú if 0 < x < 11
f(x) = í
ïî
12
if 11£ x £ 24
6) Texting Plans: Texting from T-Mobile costs $.15 per text with no plan. In addition, they offer three
other texting plans, shown to the right, that include a certain number of texts with additional texts
over costing $.15 per text.
a. Write the function rules for each where x is the number of texts and f(x) is the total monthly cost.
i.
No plan –
ii.
400 text plan–
iii.
1000 text plan–
iv.
Unlimited text plan–
b. My daughter uses approximately 1200 texts per month. How much
would this cost me under each plan?
i. No plan
ii.
400 text plan
iii.
1000 text plan
iv.
Unlimited text plan
c. I utilize about 90 texts per month. Which plan should I purchase for my phone?
d. Give the interval number of texts that would make each of these plans
the best one to purchase (this information would be good to give to our
sales people when they are advising customers on which plan to
purchase).
v.
No plan
vi.
400 text plan
vii.
1000 text plan
viii.
Unlimited text plan
Solution: By Heather period 2
ì
ï 5
if
ï
7) Let f(x) = í x +1 if
ï 1
ï - x if
î 3
a) Graph f(x).
b) Evaluate:
x £ -3
-3 < x < 0
x³0
f(8) = ____
f(-2.5) = _____
8. Write a piecewise function for the following graph.
ì 3
if
ïï
f(x) = í x 2
if
ï -x +1 if
ïî
x < -1
-1£ x £ 2,x ¹ 0
x>2
9. Graph 2x + 3y > 12 in the coordinate plane.
Solution by Lindsay & Kathy
10. A curbside recycling service will remove up to 100 pounds of plastic bottles and paper products each
week. They charge $0.25 per pound of plastic and $0.75 per pound for paper products.
a) Write an inequality that describes all possible combinations of the number of pounds of each kind
of product that can be included in the curbside service.
b) Write an inequality that describes the charge.
c) Graph each inequality.
d) Compare the two graphs.
11. Explain if each relation below is a function. Describe the domain and range. If a relation is a function, is it
one-to-one? Explain.
a)
b)
c)
d)
f(x) = 2x + 1