Piecewise Functions and Real-World Applications

A piecewise function is defined as follows:
f(x) = {x+2; if x < 0 and x² if x ≥ 0}
What if f(-3)?

• -1

• 9

• -5

• 1

Answer explanation

To find f(-3), we use the first part of the piecewise function since -3 < 0. Thus, f(-3) = -3 + 2 = -1. Therefore, the correct answer is -1.

In a cell phone plan, the monthly cost C(m) in dollars for m minutes is given by:
C(m) = {40 ;if 0 ≤ m ≤ 500 and 40+0.10(m−500) ;if m > 500}

What would be the cost for using 700 minutes?

• $40

• $60

• $70

• $20

Answer explanation

For 700 minutes, use the second part of the cost function: C(m) = 40 + 0.10(m−500). Plugging in m = 700 gives C(700) = 40 + 0.10(700−500) = 40 + 20 = $70.

Which of the following real-world scenarios would best be modeled by a piecewise function?

• The height of a growing plant over time

• The cost of parking where the first hour is free and each additional hour has a fixed rate

• The speed of a car traveling at a constant velocity

• The area of a square as its side length increases

Answer explanation

The cost of parking is a piecewise function because it has different rates for different time intervals: the first hour is free, and subsequent hours have a fixed charge, making it a clear example of a piecewise scenario.

A piecewise function h(x) equals 2x and equals 6 when x ≥ 3.

What is h(4)?

• 8

• 6

• 12

• 2

Answer explanation

To find h(4), we check the piecewise function. Since 4 ≥ 3, we use the second part of the function, which states h(x) = 6. Therefore, h(4) = 6.

At what x-value(s) is this piecewise function discontinuous?

g(x) = { x + 1​ ; if x​ < 2 and 2x−3​ ; if x ≥ 2​}

• x = 0

• The function is continuous everywhere.

• x = 1

• x = 2

Answer explanation

The function g(x) is defined piecewise. At x = 2, the left limit (2 + 1 = 3) does not equal the right limit (2*2 - 3 = 1), indicating a discontinuity. Thus, the function is discontinuous at x = 2.

A theater charges $12 for each adult ticket and $8 for each child ticket. If a group purchases a tickets and c children tickets, which piecewise function represents the total cost if there's a 20% discount when the total exceeds $100?

• C = {12a + 8c ;

  if 12a + 8c ≤ 100 

  and

  0.8(12a + 8c)​;

  if 12a + 8c​ > 100​}
  

• C = {0.8(12a + 8c);

  if 12a + 8c ≤ 100

  and

  12a + 8c; if 12a + 8c > 100}

• C = 12a + 8c

Answer explanation

The correct choice is the first one, as it accurately represents the total cost with a piecewise function: full price if total is $100 or less, and 20% discount if it exceeds $100.

• All real numbers except 0

• All real numbers greater than or equal to 0

• All real numbers less than 0

• All real numbers

Answer explanation

The function is defined for all real numbers: square root of x for x is greater than or equal to 0 and -x for x < 0. Therefore, the domain includes all real numbers.

A company's profit P(x) in thousands of dollars is given by:
P(x) = {−2x + 10 ; if 0 ≤ x ≤ 5  and x − 7 ; if x > 5
Where x is the number of units produced in thousands. What is the profit when 5000 units are produced?

• 0

• -2

• 2

• 5

Answer explanation

To find the profit for 5000 units (x = 5), use P(x) = -2x + 10. P(5) = -2(5) + 10 = 0. Thus, the profit when 5000 units are produced is 0.

Which of the following represents a continuous piecewise function at x = 2?

• f(x) = {x + 1 ; if x < 2

  and

  x + 1 ; if x ≥ 2}
  

• f(x) = {x + 1​ ; if x​ < 2 

  and

  2x−1​ ; if x ≥ 2​}
  

• f(x) = x² ; if x < 2

  and

  4 ; if x ≥ 2​}
  
  

• f(x) = {2x​ ; if x​ < 2 

  and

  x + 2 ; if x ≥ 2​}
  

Answer explanation

The function f(x) = {x + 1 ; if x < 2 and x + 1 ; if x ≥ 2} is continuous at x = 2, as both pieces equal 3 at that point, ensuring no jump or break in the function.

In a taxi fare system, the cost C(m) for m miles is:
C(m) = {3 + 2m ; if 0 ≤ m ≤ 10  and 23 + 1.5m ; if m > 10}
What is the cost for a 15-mile trip?

• $45.50

• $33

• $38

• $45

Answer explanation

For a 15-mile trip, use the second formula: C(m) = 23 + 1.5m. Plugging in m = 15 gives C(15) = 23 + 1.5(15) = 23 + 22.5 = 45. Thus, the cost is $45.