1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 4.5: Combining Functions

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Objectives o Combining functions arithmetically. o Composing functions. o Decomposing functions. o Interlude: recursive graphics.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Combining Functions Arithmetically Addition, Subtraction, Multiplication and Division of Functions 1. 2. 3. 4. The domain of each of these new functions consists of the common elements (or the intersection of elements) of the domains of f and g individually.

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: Combining Functions Arithmetically Given that solve: a. Remember that Continued on the next slide…

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: Combining Functions Arithmetically (cont.) Given that solve: b. Remember that

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 2: Combining Functions Arithmetically Given that find a. and b. a. b. Remember that

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 3: Combining Functions Arithmetically Based on the graphs of f and g below, determine the domain of and evaluate We can observe that g is 0 when x = –2 and x = 2. The domain of both f and g individually is the entire set of real numbers, so the domain of is Also based on the graphs it appears that, so

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Composing Functions Let f and g be two functions. The composition of f and g, denoted, is the function defined by. The domain of consists of all x in the domain of g for which g(x) is in turn in the domain of f. The function is read “f composed with g,” or “f of g.”

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Composing Functions Caution! Note that the order of f and g is important. In general, we can expect the function to be different from the function. In formal terms, the composition of two functions, unlike the sum and product of two functions, is not commutative.

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Composing Functions The diagram below is a schematic of the composition of two functions. The ovals represent sets, with the leftmost oval being the domain of the function g. The arrows indicate the element that x is associated with by the various functions.

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: Composing Functions Given, find: a. First, we will find g(6) by replacing x with 6 in g(x). Next, we know that f composed with g can also be written. Since we already evaluated g(6), we can insert the answer to get f(11). Continued on the next slide…

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: Composing Functions (cont.) Given, find: b. Again, we know by definition that. Note: since we solved for the variable x we should be able to plug 6 into x and get the same answer as in part a. Verify this.

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 5: Composing Functions Let. Simplify the compositions and find the domains for: a. Domain: b. Domain: Note: Note: only non-negative numbers can be plugged into g. Thus, the domain is all positive real numbers. The domain of must be any x such that since is under a radical.

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Decomposing Functions Often functions can be best understood by recognizing them as a composition of two or more simpler functions. For example, the function can be thought of as the composition of two or more functions. Note: if then:

15. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Decomposing Functions Ex: The function can be written as a composition of functions in many different ways. Some of the decompositions of f(x) are shown below: a. b. c.

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 6: Decomposing Functions Decompose the function into: a. a composition of two functions b. a composition of three functions Note: These are NOT the only possible solutions for the decompositions of f(x)!

17. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Interlude: Recursive Graphics Recursion Recursion refers to using the output of a function as its input and repeating this process a certain number of times. In other words, recursion is the composition of a function with itself some number of times. Some notations: If f is a function, is used in this context to stand for, or (not !) Similarly, stands for, or, and so on. The functions,,… are called iterates of f, with being the n ͭ ͪ iterate of f.