COMPOSITE AND INVERSE FUNCTIONS 2.4 COMPOSITE AND INVERSE FUNCTIONS

Composition of Functions

Composition of Functions For two functions f(t) and g(t), the function f(g(t)) is said to be a composition of f with g. The function f(g(t)) is defined by using the output of the function g as the input to f.

Composition of Functions Example 3 (a) Let f(x) = 2x + 1 and g(x) = x2 − 3. (a) Calculate f(g(3)) and g(f(3)). Solution (a) We want f(g(3)). We start by evaluating g(3). The formula for g gives g(3) = 32 − 3 = 6, so f(g(3)) = f(6). The formula for f gives f(6) = 2・6 + 1 = 13, so f(g(3)) = 13. To calculate g(f(3)), we have g(f(3)) = g(7), because f(3) = 7 Then g(7) = 72− 3 = 46, so g(f(3)) = 46 Notice that, f(g(3)) ≠ g(f(3)).

Composition of Functions Example 3 (b) Let f(x) = 2x + 1 and g(x) = x2 − 3. (b) Find formulas for f(g(x)) and g(f(x)). Solution (b) In general, the functions f(g(x)) and g(f(x)) are different: f(g(x)) = f(x2 – 3) = 2ˑ (x2 – 3) + 1 = 2x2 – 6 + 1 = 2x2 – 5 g(f(x)) = g(2x + 1) = (2x + 1)2 – 3 = 4x2 + 4x + 1 – 3 = 4x2 + 4x – 2

Inverse Functions The roles of a function’s input and output can sometimes be reversed. For example, the population, P, of birds on an island is given, in thousands, by P = f(t), where t is the number of years since 2007. In this function, t is the input and P is the output. Knowing the population enables us to calculate the year. Thus we can define a new function, t = g(P), which tells us the value of t given the value of P instead of the other way round. For this function, P is the input and t is the output. The functions f and g are called inverses of each other. A function which has an inverse is said to be invertible

Inverse Function Notation If we want to emphasize that g is the inverse of f, we call it f−1 (read “f-inverse”). To express the fact that the population of birds, P, is a function of time, t, we write P = f(t). To express the fact that the time t is also determined by P, so that t is a function of P, we write t = f −1(P). The symbol f−1 is used to represent the function that gives the output t for a given input P. Warning: The −1 which appears in the symbol f−1 for the inverse function is not an exponent.

Finding a Formula for the Inverse Function Example 6 The cricket function, which gives temperature, T , in terms of chirp rate, R, is T = f(R) = ¼・R + 40. Find a formula for the inverse function, R = f−1(T ). Solution The inverse function gives the chirp rate in terms of the temperature, so we solve the following equation for R: T = ¼ ・R + 40, giving T − 40 = ¼ ・R , R = 4(T − 40). Thus, R = f−1(T ) = 4(T − 40).

Domain and Range of an Inverse Function Functions that possess inverses have a one-to-one correspondence between elements in their domain and elements in their range. The input values of the inverse function f −1 are the output values of the function f. Thus, the domain of f −1 is the range of f. Similarly, the domain of f is the range of f -1 .