1. NEW FUNCTIONS FROM OLD 1

2. 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

3. 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

4. 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

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

6. 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

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

8. 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

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

10. 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

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

12. 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

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

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

15. 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

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

17.  The domain of is: COMPOSITION Example 7 a

18.  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

19.  The domain of is. COMPOSITION Example 7 c

20.  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

21. 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

22. Find if, and. Example 8 COMPOSITION

23. 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

24. 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

25.  Then, COMPOSITION Example 9