# piecewise functions by sofarsogood

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```									                                             Piecewise Functions

For problems 1-3, evaluate each piecewise function at the given values of the independent variable.

6 x − 1    if x < 0
1. f ( x) =                                       a. f (−3)   b. f (0)       c. f (4)
7 x + 3    if x ≥ 0

 x2 − 9
           if x ≤ −1
2. f ( x) =  x + 2                                a. f (−3)   b. f (1)
6           if x > −1



2 + x     if x < −4

3. f ( x) = − x       if − 4 ≤ x ≤ 2              a. f (2)    b. f (3)
1
 x        if x > 2
3

4. When a diabetic takes long-acting insulin, the insulin reaches its peak effect on the blood sugar
level in about 3 hours. This effect remains fairly constant for 5 hours, then declines, and is very low
until the next injection. In a typical patient, the level of insulin might be modeled by the following
function.

40t + 100      if 0 ≤ t ≤ 3
220            if 3 < t ≤ 8

f (t ) = 
−80t + 860     if 8 < t ≤ 10
60
               if 10 < t ≤ 24

Here, f (t ) represents the blood sugar level at time t hours after the time of the injection. If a patient
takes insulin at 6 am, find the blood sugar level at each of the following times.

a. 7 am                    b. 11 am              c. 3 pm              d. 5 pm

For problems 5-10, graph each piecewise function.

 x + 3 if x < −1                                              x −1   if x ≤ 3
5. f ( x) =                                                   6. f ( x) = 
2 x − 1 if x ≥ −1                                             2      if x > 3
−1      if x < 0                                           4 − x if x ≤ 2
7. f ( x) =                                                8. f ( x) = 
 x − 3 if x ≥ 0                                            3x − 6 if x > 2

−2 x     if x < −1                                          2 + x   if x < −2
                                                            
9. f ( x) = 3 x − 1 if − 1 ≤ x ≤ 2                         10. f ( x) = − x     if − 2 ≤ x ≤ 1
 1                                                          0       if x > 1
− x if x > 2                                                
 2

For problems 11-13, give the piecewise function that each graph represents.
11.                              12.                               13.

For problems 14-17, write a piecewise function that describes each situation.

14. For a cellular phone billing plan, \$50 per month buys 400 minutes or less. Additional time costs \$0.30 per
minute. Let the monthly cost C(x) be a function of the time x.

15. For a cellular phone billing plan, \$60 per month buys 450 minutes or less. Additional time costs \$0.35 per
minute. Let the monthly cost C(x) be a function of the time x.

16. Income tax is 5% on the first \$50,000 of income or less, and 8% on any income in excess of \$50,000. Let
the tax T(x) be a function of the income x.

17. In Missouri, income tax is 3.5% on the first \$9,000 of income or less, and 6% on any income in excess of
\$9,000. Let the tax T(x) be a function of the income x.
Solutions – Piecewise Functions

1a. f (−3) = −19               1b. f (0) = 3                1c. f (4) = 31

2a. f (−3) = 0                 2b. f (1) = 6

3a. f (2) = −2                 3b. f (3) = 1
4a. 140 units   4b. 220 units         4c. 140 units   4d. 60 units

5.                              6.

7.                              8.

9.                              10.
1
 x+2        if x < 4
11. f ( x) =  4
1
            if x ≥ 4

−2           if x ≤ −3

12. f ( x) =  −1
 3 x+3

if x > −3

 −2 x − 1   if x ≤ 0                                         −2 x − 1   if x < 0
13. f ( x) =                                        or          f ( x) = 
3x − 1      if x > 0                                        3x − 1      if x ≥ 0

50               if 0 ≤ x ≤ 400                           50            if 0 ≤ x ≤ 400
14. C ( x) =                                        or        C ( x) = 
50 + 0.30( x − 400) if x > 400                            0.30 x − 70       if x > 400

60               if 0 ≤ x ≤ 450                             60          if 0 ≤ x ≤ 450
15. C ( x) =                                        or          C ( x) = 
60 + 0.35( x − 450) if x > 450                              0.35 x − 97.5 if x > 450

0.05 x                        if 0 ≤ x ≤ 50000
T ( x) =                                                                    0.05 x     if 0 ≤ x ≤ 50000
16.           0.05(50000) + 0.08( x − 50000) if x > 50000     or       T ( x) = 
0.08 x − 1500 if x > 50000

0.035 x                     if 0 ≤ x ≤ 9000                         0.035 x      if 0 ≤ x ≤ 9000
17. T ( x) =                                                   or       T ( x) = 
0.035(9000) + 0.06( x − 9000) if x > 9000                           0.06 x − 225     if x > 9000

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