Invertible Functions Worksheets Grade 12

1. Mere is a marine biologist studying the population of a certain fish species in a lake. She models the fish population (in thousands) over time using the function f(t) = t^3 - 15t^2 + 63t, where t represents time in years since the study began. The population initially increases, then decreases, and then increases again. To create an inverse function that can predict the time when the population first reaches a given value during the initial increasing phase, Mere needs to restrict the domain to the interval where the function is strictly increasing and includes t = 0. Determine the largest possible domain of the form [0, a] that makes f(t) invertible.
2. Sophia is an electrical engineer analyzing the voltage output of a prototype circuit over time. The voltage is modeled by the function V(t) = t^3 - 6t^2 + 9t + 1, where t is time in seconds and V(t) is in volts. To design a feedback control system that requires a one-to-one relationship between time and voltage, Sophia must restrict the domain to an interval where the function is strictly decreasing. Determine the largest possible interval of the form [a, b] on which V(t) is strictly decreasing and therefore invertible.
3. f(x) = (x - 4)² is not invertible. Find the domain restriction x ≥ k that makes it invertible.

…and 3 more problems

1. Consider the function f(x) = (x - 6)^2 - 1, which represents a parabola opening upward with its vertex at (6, -1). Visualize this U-shaped curve on a coordinate plane. The function fails the horizontal line test over its entire domain because a horizontal line can intersect the parabola at two points. Determine the largest possible restriction of the domain to the right of the vertex (including the vertex) so that the resulting function is invertible. Express your answer using interval notation.
2. f(x) = (x - 6)² + 2; restrict domain to make invertible
3. Noah is a pharmaceutical researcher modeling the concentration of a new antibiotic in a patient's bloodstream. The concentration, in mg/L, is given by the function C(t) = 12t^2 - t^3, where t is the time in hours after administration, for 0 ≤ t ≤ 12. To use the model for predicting the exact time when a specific concentration occurs, Noah needs to restrict the domain to a smaller interval where the function is one-to-one (invertible). Determine the largest possible interval of the form [a, b] within [0, 12] where C(t) is strictly decreasing, making it invertible.

…and 4 more problems

1. Sophia is an astrophysicist modeling the gravitational potential energy of a satellite orbiting a planet. The energy (in gigajoules) as a function of orbital radius r (in thousands of kilometers) is given by E(r) = 2r^3 - 30r^2 + 126r - 10. For her analysis of orbital stability, she needs to restrict the domain to an interval where the energy function is strictly decreasing, ensuring the function is invertible. Determine the largest possible interval of the form [a, b] where E(r) is strictly decreasing and therefore invertible.
2. Sophia is a pharmaceutical researcher modeling the rate at which a new antibiotic is absorbed into bacterial cells. The absorption rate (in micrograms per minute) is given by the function f(x) = x^3 - 15x^2 + 63x - 49, where x represents the time in minutes after the antibiotic is introduced. To analyze the period when the absorption rate is strictly decreasing, Sophia needs to restrict the domain of f(x) to make it invertible. Determine the largest possible interval of the form [a, b] where f(x) is strictly decreasing and therefore invertible.
3. A civil engineer is designing a parabolic arch bridge that follows the equation f(x) = -x² + 8x - 12, where x represents the horizontal distance from the left support in meters. To ensure the bridge's structural analysis can be properly modeled, she needs to restrict the domain to make the function invertible while maintaining the portion where the arch is increasing. Determine the largest possible interval of the form [a, b] where the function is strictly increasing and therefore invertible.

…and 3 more problems