Composite Functions Worksheets Grade 12

1. Emma is analyzing the motion of a particle in a physics experiment. The particle's position is given by f(x) = sqrt(x - 4), where x represents time in seconds. She applies a transformation g(x) = 1/(x - 2) to model how the particle's energy changes with position. To understand how energy depends on time, Emma needs to find the domain of the composite function (g∘f)(x). What is the domain of this composite function?
2. Aroha is a geotechnical engineer analyzing the stability of a rock formation. The depth D(t) of a borehole in meters, t hours after drilling begins, is modeled by D(t) = sqrt(2t - 16). The pressure P(D) in kilopascals at depth D is given by P(D) = 1 / (D - 13). To ensure the drilling equipment is safe, Aroha needs to find the domain of the composite function P(D(t)) that describes pressure as a function of time. What is the domain of P∘D?
3. A marine biologist is studying the temperature-dependent growth of a specific algae species in a controlled environment. The water temperature T in degrees Celsius is modeled by T(t) = √(t + 4), where t is time in days. The algae growth rate G is given by G(T) = 1/(T - 2), which represents how quickly the algae population increases per day at a given temperature. To understand how the growth rate changes over time, the biologist needs to find the domain of the composite function G(T(t)). What is the domain of G∘T?

…and 3 more problems

1. Charlotte is a pharmacist formulating a slow-release medication. The concentration of the drug in a patient's bloodstream t hours after administration is modeled by C(t) = ln(t - 9), where C is in milligrams per liter. The body's response rate R to the drug concentration is given by R(C) = 1 / (C - 12), measured in arbitrary units. To ensure patient safety, Charlotte needs to determine the domain of the composite function R(C(t)) that describes the response rate as a function of time. What is the domain of R∘C?
2. Sophia is modeling the spread of a beneficial fungus through a forest's root system. The area A(t) in square meters covered by the fungus after t weeks is given by A(t) = ln(t - 7). The nutrient absorption rate N (in grams per week) depends on the area covered and is modeled by N(A) = 1 / (A - 9). Sophia wants to understand how the nutrient absorption rate changes over time by studying the composite function N(A(t)). Determine the domain of this composite function, considering the real-world context that time t is measured in weeks.
3. A marine biologist is studying the population growth of dolphins in a protected bay. The dolphin population P(t) after t years is modeled by P(t) = 500e^(0.03t). The available food supply F in kilograms is given by F(p) = 2000 - 2p, where p is the dolphin population. Determine the domain of the composite function F(P(t)) that represents the food supply as a function of time, considering the biological constraints that both population and food supply must remain non-negative.

…and 3 more problems

1. A marine biologist is studying the temperature-dependent growth of a coral species. The water temperature T in degrees Celsius is modeled by T(t) = 20 + 5sin(πt/6), where t is time in months. The coral growth rate G is given by G(T) = ln(T - 15). To understand how coral growth varies over time, the biologist needs to find the domain of the composite function G(T(t)). What is the domain of this composite function?
2. Given f(x) = √(x - 6) and g(x) = 1/(x² - 121), find the domain of (g∘f)(x) in interval notation.
3. Given f(x) = √(x - 12) and g(x) = 1/(x² - 81), find the domain of (g∘f)(x) in interval notation.

…and 4 more problems