Piecewise Functions Worksheets Grade 12

1. Kaia is a civil engineer designing a water drainage system for a new housing development. The water flow rate (in liters per second) through a drainage pipe is modeled by the piecewise function f(x) = { 3x + 1 for -3 ≤ x < 1; -x² + 5x - 1 for 1 ≤ x ≤ 5 }, where x represents the time in hours after a storm begins. Kaia needs to evaluate the flow rate at specific times to check for overflow risks. Evaluate f(-3), f(1), f(3), and f(5). Then, describe the behavior of the graph at x = 1, specifically whether there is a jump, break, or continuity, and identify any other discontinuities in the domain.
2. Sophia is analyzing a step function defined as g(x) = { 12 for -7 ≤ x < -4; 8 for -4 ≤ x < 1; -3 for 1 ≤ x ≤ 7 }. On a coordinate plane, graph this function and evaluate g(-5), g(-4), g(0), and g(7).
3. f(x) = {x^2 - 11 if x < -4; 3x + 10 if -4 ≤ x < 2; floor(x) + 7 if x ≥ 2}. Evaluate f(-5), f(-4), f(1), f(2), f(6)

…and 4 more problems

1. A circular oil spill is expanding such that its radius increases at a rate of 2 meters per hour. At a particular moment, the radius is 50 meters. Using calculus, determine how fast the area of the oil spill is increasing at that moment. (Use π in your answer.)
2. Isabella is a structural engineer designing a support beam for a pedestrian bridge. The beam's height above the ground (in meters) along its length is modeled by the piecewise function: f(x) = { 3x + 2 for 0 ≤ x < 4; -x² + 12x - 18 for 4 ≤ x ≤ 8 } where x is the horizontal distance in meters from the left support. Isabella needs to verify the beam's profile for safety. Evaluate f(0), f(4), and f(7), then determine the height of the beam at x = 4 (the connection point between the two pieces) and identify whether the beam has a jump discontinuity at that point.
3. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream using the piecewise function C(t) = { 2t^2 - 4t + 3 for 0 ≤ t < 2; 5e^(-0.5(t-2)) + 3 for t ≥ 2 }, where t is hours after administration and C(t) is measured in mg/L. The drug is considered effective when concentration exceeds 4 mg/L and safe when concentration is below 8 mg/L. Determine the time intervals during which the drug is both effective and safe.

…and 2 more problems

1. Liam is designing a custom skateboard ramp that has a piecewise function for its cross-sectional height. The height function h(x) in meters is defined as: h(x) = { 0.5x + 1 for 0 ≤ x ≤ 2; -x² + 5x - 2 for 2 < x ≤ 4; 2 for 4 < x ≤ 6 }. Liam needs to determine if the ramp has any discontinuities in its height profile. At what x-value does a discontinuity occur, and what type of discontinuity is it?
2. Liam is modeling the temperature in a chemical reaction chamber using a piecewise function. The temperature T(t) in degrees Celsius at time t minutes is defined as: T(t) = { 2t^2 - 8t + 10 for 0 ≤ t < 3; -t^2 + 10t - 11 for 3 ≤ t ≤ 7 }. Liam needs to determine the exact time when the temperature reaches its minimum value during the first 7 minutes of the reaction. At what time does this minimum temperature occur?
3. Charlotte is a financial analyst evaluating a special tax structure for a new investment fund. The fund's effective tax rate T(x) as a percentage of profit (in thousands of dollars) is defined by the piecewise function: T(x) = { 0.12x + 2 for 0 ≤ x < 7; 0.27x - 0.73 for 7 ≤ x ≤ 12 }, where x is the profit in thousands of dollars. Charlotte needs to determine the effective tax rate for profits of $2,000, $7,000, and $12,000. She also needs to identify any discontinuity in the tax rate at the boundary x = 7. What are T(2), T(7), and T(12), and is there a jump discontinuity at x = 7?

…and 3 more problems