Final answer:
To find the inverse function where f-1(P) = p, we use the formula for the power of a lens P = 1/f, resulting in the inverse function f-1(P) = 1/P. Understanding inverse relationships in mathematics, like inversely proportional variables or negative exponents, helps in deriving formulae for inverse functions.
Step-by-step explanation:
To find a formula for the inverse function where f-1(P) = p, we need to consider the definition of power of a lens. The power P of a lens is defined as the inverse of its focal length f, so the formula is P = 1/f. To find the inverse of this function, we interchange P and f, and solve for f, which gives us f = 1/P or equivalently f-1(P) = 1/P.
Understanding the relationship between variables that are inversely proportional, such as pressure (P) and volume (V), helps us grasp the concept of inverse functions in various contexts. For example, in the ideal gas law PV = nRT, P is inversely proportional to V when n, R, and T are held constant, indicating that as V increases, P decreases and vice versa. Similarly, understanding exponents and their inverses aids in working with powers and roots, such as knowing that x-n = 1/xn and the inverse of a square is a square root.