Invertible Functions Worksheets Grade 12

1. f(x) = (x - 4)² is not invertible. Find the domain restriction x ≥ k that makes it invertible.
2. f(x) = (x - 7)² + 2 is not invertible. Find the domain restriction x ≥ a that makes it invertible.
3. f(x) = (x - 6)² + 1 is not invertible. Find the domain restriction x ≥ a that makes it invertible.

…and 5 more problems

1. Noah is an environmental engineer modeling the temperature of a chemical reaction over time. The temperature in degrees Celsius is given by the function f(x) = x^3 - 12x^2 + 36x + 1, where x represents time in minutes. To analyze the reaction's behavior during a specific phase, Noah needs to restrict the domain to an interval where the function is strictly decreasing, making it invertible. Determine the largest possible interval of the form [a, b] where f(x) is strictly decreasing and therefore invertible.
2. A solid is formed by rotating the region bounded by the curve y = x³, the x-axis, and the vertical line x = 1 about the y-axis. Using the method of cylindrical shells, set up the integral expression for the volume of this solid. Describe the visual geometric elements: the cubic curve, the bounded region under the curve from x=0 to x=1, and the rotation around the y-axis creating a three-dimensional volume.
3. A pharmaceutical company is modeling the concentration of a new drug in the bloodstream over time using the function C(t) = (t² - 9)/(t - 3), where t represents hours after administration. The function is undefined at t = 3 due to division by zero. To make the function continuous and invertible for their pharmacokinetic analysis, they need to restrict the domain by removing the problematic point. What is the maximum domain on which this function becomes both continuous and invertible?

…and 3 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 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. 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.

…and 3 more problems