Function Inverses Worksheets Grade 12

1. Aroha is a materials engineer testing the thermal expansion of a new alloy. She models the length of a metal rod (in centimeters) as a function of temperature T (in Celsius) by L(T) = 3T - 7. She suspects that the inverse function, which gives the temperature needed to achieve a certain length, is T(L) = (L + 7)/3. Verify by composition whether L(T) and T(L) are indeed inverse functions for all values in their domains.
2. Given f(x) = 5x - 12 and g(x) = (x + 12)/5, verify f(g(23)) = ?
3. Isabella is a materials scientist modeling the expansion of a metal rod under heat. The length L (in cm) of the rod at temperature T (in degrees Celsius) is given by L(T) = (3/5)T + 14. Her colleague, Mason, proposes an inverse function T(L) = (5/3)(L - 14) to determine the temperature from a given length. Using function composition, verify whether L(T) and T(L) are indeed inverse functions by computing L(T(L)) and T(L(T)) and checking if both simplify to the original variable.

…and 4 more problems

1. Matiu is an engineer designing a control system for a robotic arm. The arm's position in centimeters is given by the function f(x) = 4x - 12, where x is the voltage input in volts. The control system uses a calibration function g(x) = (x + 12)/4 to convert the sensor readings back to voltage. Matiu needs to verify that these two functions are inverses of each other by checking their compositions. Compute f(g(x)) and g(f(x)) and determine if they both equal x.
2. Liam is designing a new encryption algorithm that relies on a pair of functions to encode and decode messages. The encoding function is f(x) = (3x - 7) / 5, and the decoding function is g(x) = (5x + 7) / 3. To verify his algorithm works correctly, Liam must confirm that f and g are inverse functions. By computing f(g(x)) and g(f(x)), determine if these two functions are indeed inverses of each other.
3. f(x) = 2x + 3 and g(x) = (x - 3)/2, find f(g(5)) = ?

…and 3 more problems

1. A function f(x) = (x - 2)³ + 1 is graphed on a coordinate plane. Its inverse function f⁻¹(x) is also graphed, reflected across the line y = x. If you start at the point (3, 2) on f(x) and follow this path: move horizontally to the line y = x, then vertically to f⁻¹(x), what are the coordinates of the final point reached on f⁻¹(x)?
2. Kaia is a hydrologist modeling the flow rate of a river after a storm. The flow rate in cubic meters per second is given by the function f(x) = 4x - 9, where x is the water level in meters above flood stage. Her colleague Tane proposes that the inverse function is g(x) = (x + 9)/4. Using function composition, verify whether f(x) and g(x) are inverse functions by computing f(g(x)) and g(f(x)).
3. Noah is a financial analyst studying the relationship between two economic models. The first model describes the cost of manufacturing x units of a product as f(x) = (6x - 1) / 5. The second model describes the number of units produced based on a budget of y dollars as g(y) = (5y + 1) / 6. Noah wants to verify that these two functions are inverses of each other, meaning that applying one after the other returns the original input. Use function composition to determine if f(g(x)) = x and g(f(x)) = x hold for all values of x.

…and 4 more problems