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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 .

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2 4

2.4

COMPOSITE AND INVERSE FUNCTIONS


2 4

Composition of 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 functions1

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 functions2

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

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 inversesof each other. A function which has an inverse is said to be invertible


Inverse function notation

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−1is 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

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

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 −1is the range of f.

  • Similarly, the domain of f is the range of f -1 .


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