Inverse Functions Worksheets Grade 12

1. A biologist is modeling the population growth of a rare species using the function P(t) = 500e^(0.03t), where t represents time in years and P(t) represents the population size. To determine how long it will take for the population to reach a specific target size, the biologist needs to find the inverse function. Find the inverse function t(P) that gives the time required to reach population P.
2. A biomedical company is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (2t + 3)/(t - 1), where C represents concentration in mg/L and t represents time in hours. To determine when the concentration reaches a specific level, they need to find the inverse function. What is the inverse function t(C) that gives the time when the concentration is C mg/L?
3. A function is represented graphically as a cubic curve with inflection point at (1, 2) and passing through points (0, 3) and (2, 1). The function has the form f(x) = a(x - h)³ + k. Find the algebraic expression for its inverse function f⁻¹(x).

…and 3 more problems

1. Aroha is a climate scientist studying the relationship between ocean temperature and depth in a specific trench. The temperature T(d) in degrees Celsius at a depth d kilometers is modeled by the rational function T(d) = (9d + 7)/(d - 3), where d > 3. To predict the depth at which a specific temperature occurs, Aroha needs to find the inverse function. Determine the inverse function d(T) that gives the depth in kilometers when the temperature is T degrees Celsius.
2. f(x) = (13x + 9)/(11x - 8). Find f⁻¹(x) = ?
3. Olivia is a materials scientist studying the thermal expansion of a new ceramic composite. The length L(t) of a sample in millimeters at temperature t (in degrees Celsius) is modeled by the function L(t) = (3t + 1) / (t - 3). To predict the temperature required to achieve a specific length for manufacturing tolerances, Olivia needs the inverse function that expresses temperature as a function of length. Find the inverse function t(L) that gives the temperature in degrees Celsius when the length is L millimeters.

…and 3 more problems

1. A graph on a coordinate plane shows a curve that is symmetric about the line y = x. The function f(x) passes through the points (1, -3), (3, 1), and (5, 5). Given that f(x) = ax^3 + bx^2 + cx + d, find f⁻¹(5).
2. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (2t + 3)/(t - 1), where C is the concentration in milligrams per liter and t is time in hours after administration. The researchers need to determine at what time the concentration reaches a specific level. Find the inverse function C⁻¹(c) that would allow them to calculate the time when the concentration equals any given value c.
3. Mere is an agricultural scientist modeling the rate of water absorption by a special type of soil. She uses the function A(t) = (3t + 13)/(t - 5), where A represents the absorption rate in liters per hour and t represents time in hours since the start of an irrigation experiment. To predict when the absorption rate will reach a specific target level, Mere needs to find the inverse function. What is the inverse function A⁻¹(x) that gives the time when the absorption rate is x liters per hour?

…and 4 more problems