NEW FUNCTIONS FROM OLD

1. 1 NEW FUNCTIONS FROM OLD

2. FUNCTIONS AND MODELS 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.

3. COMBINATIONS OF FUNCTIONS 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.

4. SUM AND DIFFERENCE 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.

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

6. PRODUCT AND QUOTIENT 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

7. PRODUCT AND QUOTIENT For instance, if f(x) = x2 and g(x) = x - 1, then the domain of the rational function is , or

8. COMBINATIONS 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:

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

10. COMPOSITION 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”).

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

12. 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.

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

14. COMPOSITION Example 6 If f(x) = x2 and g(x) = x - 3, find the composite functions and . • We have:

15. COMPOSITION Note 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 firstsubtracts 3 and thensquares; is the function that firstsquares and thensubtracts 3.

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

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

18. COMPOSITION Example 7 b • 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].

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

20. COMPOSITION Example 7 d • 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].

21. 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 fas follows:

22. COMPOSITION Example 8 Find if , and .

23. 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.

24. COMPOSITION Example 9 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:

25. COMPOSITION Example 9 • Then,