Section 3.4: Piecewise-defined functions
#1-6: Find the indicated value for each function.
1)
𝑓(𝑥) = {
3𝑥, 𝑖𝑓 𝑥 < 0
2𝑥 + 1, 𝑖𝑓 𝑥 ≥ 0
a) f(-5)
b) f(0)
c) f(2)
𝑥 − 5, 𝑖𝑓 𝑥 ≤ 5
2) 𝑓(𝑥) = {
2𝑥 − 4, 𝑖𝑓 𝑥 > 5
a) f(0)
b) f(5)
c) f(6)
𝑥 − 5, 𝑖𝑓 𝑥 < −1
𝑥, 𝑖𝑓 − 1 ≤ 𝑥 ≤ 2
3) 𝑔(𝑥) = {
𝑥 + 2, 𝑖𝑓 𝑥 > 2
a) g(-1)
b) g(2)
c) g(0)
2𝑥, 𝑖𝑓 𝑥 ≤ 0
4) g(x) = {𝑥, 𝑖𝑓 0 < 𝑥 ≤ 3
−5𝑥 𝑖𝑓 𝑥 > 3
a) g(0)
b) g(3)
c) g(-2)
𝑥 2 − 10, 𝑖𝑓 𝑥 < −10
5) 𝑘(𝑥) = {𝑥 2 , 𝑖𝑓 − 10 ≤ 𝑥 ≤ 10
𝑥 2 + 10. 𝑖𝑓 𝑥 > 10
a) k(-10)
b) k(11)
c) k(0)
2𝑥 2 − 3, 𝑖𝑓 𝑥 < 2
6) 𝑘(𝑥) = { 𝑥 2 , 𝑖𝑓 2 ≤ 𝑥 ≤ 4
5𝑥 − 7 𝑖𝑓 𝑥 > 4
a) k(2)
b) k(4)
c) k(5)
#7-12: sketch a graph of each function.
7)
𝑓(𝑥) = {
3𝑥, 𝑖𝑓 𝑥 < 0
2𝑥 + 1, 𝑖𝑓 𝑥 ≥ 0
𝑥 − 5, 𝑖𝑓 𝑥 ≤ 5
8) 𝑓(𝑥) = {
2𝑥 − 4, 𝑖𝑓 𝑥 > 5
𝑥 − 5, 𝑖𝑓 𝑥 < −1
𝑥, 𝑖𝑓 − 1 ≤ 𝑥 ≤ 2
9) 𝑔(𝑥) = {
𝑥 + 2, 𝑖𝑓 𝑥 > 2
2𝑥, 𝑖𝑓 𝑥 ≤ 0
10) g(x) = {𝑥 + 1, 𝑖𝑓 0 < 𝑥 ≤ 3
−5𝑥 𝑖𝑓 𝑥 > 3
𝑥 2 − 2, 𝑖𝑓 𝑥 < −1
11) 𝑘(𝑥) = {𝑥 2 , 𝑖𝑓 − 1 ≤ 𝑥 ≤ 1
𝑥 2 + 2. 𝑖𝑓 𝑥 > 1
2𝑥 2 − 3, 𝑖𝑓 𝑥 < 2
12) 𝑘(𝑥) = { 𝑥 2 , 𝑖𝑓 2 ≤ 𝑥 ≤ 4
5𝑥 − 7 𝑖𝑓 𝑥 > 4
13) In the 1995 tax form a tax rate schedule is given for people whose filing status is single. Part of the
table is shown below:
If the taxable income is over... But not over-- then the tax is... of the amount over-$0
$23,350
15%
$0
$23,350
$56,550 $3,502.50 + 28%
$23,350
$56,550
$117,950 $12,798.50 + 31%
$56,550
a. Write the defining rule for a piecewise function T(x) giving the tax owed by a person
whose taxable income is x, where x is less than $117,950.
b. Evaluate the function to find the tax owed by a single person whose taxable income in
1995 was $31,950.
14) In the 2005 tax form a tax rate schedule is given for people whose filing status is single. Part of the
table is shown below:
If the taxable income is over... But not over-- then the tax is... of the amount over-$0
$28,000
10%
$0
$28,000
$60,000
$2800 + 20%
$28,000
$60,000
$200,000
9200 + 25%
$60,000
a) Write the defining rule for a piecewise function T(x) giving the tax owed by a person
whose taxable income is x, where x is less than $200,000.
b) Evaluate the function to find the tax owed by a single person whose taxable income
in 2005 was $50,000.
15) Assume you work at a company where you are paid hourly.
You are paid $7.80 per hour for regular time (less than or equal to thirty-five hours)
and time and a half for overtime hours up to forty-five hours in one week.
If you are asked to work forty-five or more hours in one week you are paid double-time.
a) Write this information in the form of a piecewise-defined function
b) Use your formula to compute the pay for working 50 hours in a week.
16) Assume you work at a company where you are paid hourly.
You are paid $10.00 per hour for regular time (less than or equal to forty hours)
and time and a half for overtime hours up to fifty hours in one week.
If you are asked to work fifty or more hours in one week you are paid double-time.
a) Write this information in the form of a piecewise-defined function
b) Use your formula to compute the pay for working 50 hours in a week.