NEW FUNCTIONS FROM OLD 1

1.3 New Functions from Old Functions In this section, we will learn: How to obtain new functions from old functions and how to combine pairs of functions. FUNCTIONS AND MODELS

Two functions f and g can be combined to form new functions f + g, f - g, fg, and in a manner similar to the way we add, subtract, multiply, and divide real numbers. COMBINATIONS OF FUNCTIONS

The sum and difference functions are defined by: (f + g)x = f(x) + g(x) (f – g)x = f(x) – g(x)  If the domain of f is A and the domain of g is B, then the domain of f + g is the intersection.  This is because both f(x) and g(x) have to be defined. SUM AND DIFFERENCE

For example, the domain of is and the domain of is. So, the domain of is. SUM AND DIFFERENCE

Similarly, the product and quotient functions are defined by:  The domain of fg is.  However, we can’t divide by 0.  So, the domain of f/g is PRODUCT AND QUOTIENT

For instance, if f(x) = x 2 and g(x) = x - 1, then the domain of the rational function is, or PRODUCT AND QUOTIENT

There is another way of combining two functions to obtain a new function.  For example, suppose that and  Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x.  We compute this by substitution: COMBINATIONS

This procedure is called composition— because the new function is composed of the two given functions f and g. COMBINATIONS

In general, given any two functions f and g, we start with a number x in the domain of g and find its image g(x).  If this number g(x) is in the domain of f, then we can calculate the value of f(g(x)).  The result is a new function h(x) = f(g(x)) obtained by substituting g into f.  It is called the composition (or composite) of f and g.  It is denoted by (“f circle g”). COMPOSITION

Given two functions f and g, the composite function (also called the composition of f and g) is defined by: Definition COMPOSITION

The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.  In other words, is defined whenever both g(x) and f(g(x)) are defined. COMPOSITION

The figure shows how to picture in terms of machines. COMPOSITION

If f(x) = x 2 and g(x) = x - 3, find the composite functions and.  We have: Example 6 COMPOSITION

You can see from Example 6 that, in general,.  Remember, the notation means that, first, the function g is applied and, then, f is applied.  In Example 6, is the function that first subtracts 3 and then squares; is the function that first squares and then subtracts 3. COMPOSITION Note

If and, find each function and its domain. a. b. c. d. Example 7 COMPOSITION

 The domain of is: COMPOSITION Example 7 a

 For to be defined, we must have.  For to be defined, we must have, that is,, or.  Thus, we have.  So, the domain of is the closed interval [0, 4]. Example 7 b COMPOSITION

 The domain of is. COMPOSITION Example 7 c

 This expression is defined when both and.  The first inequality means.  The second is equivalent to, or, or.  Thus,, so the domain of is the closed interval [-2, 2]. Example 7 d COMPOSITION

It is possible to take the composition of three or more functions.  For instance, the composite function is found by first applying h, then g, and then f as follows: COMPOSITION

Find if, and. Example 8 COMPOSITION

So far, we have used composition to build complicated functions from simpler ones. However, in calculus, it is often useful to be able to decompose a complicated function into simpler ones—as in the following example. COMPOSITION

Given, find functions f, g, and h such that.  Since F(x) = [cos(x + 9)] 2, the formula for F states: First add 9, then take the cosine of the result, and finally square.  So, we let: Example 9 COMPOSITION

 Then, COMPOSITION Example 9