1. 1 1.6 Operations with Functions and Composition In this section, we will combine functions using the four basic operations: addition, subtraction, multiplication and division. We will also learn a new operation called composition. If f and g are functions and D is the intersection of the domains, then the following definitions can be made: Consider two functions, f(x) and g(x), where f(x)=x 2 +5x and g(x)=3x+7. Note that the domains of these two functions are all real numbers. Now, suppose we wanted to add these two functions together. “f(x) + g(x)” would be the same as “(x 2 +5x)+ (3x+7)”, or more precisely “ x 2 +8x+7” and the domain of this sum would be the intersection of the domains of the original functions. The domain of f(x)= D f : The domain of f(x)= D g : The domain of f(x) + g(x)= D f+g : Any value of x that is an element of both domains will be an element of the domain for the sum of the functions.

2. 2 1.6 Operations with Functions and Composition The procedure for finding the difference between two functions is just as easy. Keep in mind that we are working with the functions f(x)=x 2 +5x and g(x)=3x+7. The difference between the two functions would be written as “f(x) – g(x)”. f(x) – g(x)=(x 2 +5x) – (3x+7)= x 2 +5x –3x – 7 f(x) – g(x)= x 2 +2x –7 Note: Putting temporary parentheses around the two functions that we are combining is a good idea, but they are vital to obtaining a correct answer when subtracting two functions, to ensure that the subtraction sign gets distributed to each term of the second function. The domain of the sum, difference, and product of two functions are the intersection of the domains of the two functions. Procedure: To find the domain of sum, difference and product of two functions. Next Slide

3. 3 1.6 Operations with Functions and Composition Your Turn Problem #1 Solution: (a) f+g=f(x)+g(x)(b) f – g=f(x) – g(x)

4. 4 1.6 Operations with Functions and Composition Find the domains of the individual functions f(x) and g(x). Solution: D f+g : 2-3 D f+g : x≠2 in the function f(x). 2 x≥ – 3 in the function g(x). 2-3 Dg:Dg: Your Turn Problem #2

5. 5 1.6 Operations with Functions and Composition Consider the functions f(x)=x 2 +6x – 4 and g(x)=7x – 2. To find the quotient of these two functions, we find f/g. For this particular solution, the quotient cannot be simplified. To find the domain of a quotient of two functions, you must still find the intersection of the domains of the individual functions, but you must leave out any values of x that would cause the function in the denominator to equal zero. To find the domain of f/g for the functions f(x)=x 2 +6x–4 and g(x)=7x–2, we must consider the domain of both f and g. The domain of f is all real numbers, or, in interval notation,(-∞,∞). The domain of g is also all real numbers. The intersection of these two domains is all real numbers. However, we must omit from the domain of f/g any values of x that would cause the function g(x) to equal zero. Setting g(x) equal to zero and solving for x yields: 7x – 2=07x=2 The domain of f/g is all real numbers except x=2/7. In interval notation, D f/g =

6. 6 1.6 Operations with Functions and Composition Example 3. If f(x)=x 2 –2x –15 and g(x)=x 2 +5x+4, find f/g. Also specify its domain. Solution: Since the domains of the individual functions f and g are both “all real numbers”, all we need to do is find what values of x make the function in the denominator, g(x), equal zero, and then exclude those values from the domain of f/g. 0= x 2 +5x+4 x= –4x= –1 The domain of f/g is: all real numbers except x= –4 and x= –1. Your Turn Problem #3

7. 7 1.6 Operations with Functions and Composition f(5)=(5) 2 – 3=25 – 3=22 f(5)= 22 Consider the function f(x)= x 2 – 3. The domain of this function is all real numbers. f(5) can be found by replacing the “x” in “x 2 – 3” with a “5”, and simplifying. Composition “x” can also be replaced with variables or expressions that represent real numbers. f(m)=(m) 2 – 3 =m 2 – 3 f(a+b)=(a+b) 2 – 3 =a 2 +2ab+b 2 – 3 or Let’s now consider the function g(x)=x – 7. The domain of this function is all real numbers. It is possible to find f(g(x)), which is equivalent to f(x – 7). f(g(x))=f(x – 7) =(x – 7) 2 – 3 = x 2 – 14x+49 – 3 f(g(x))= x 2 – 14x+46 We have just composed two functions to produce a brand new function. Next Slide

8. 8 1.6 Operations with Functions and Composition Solution: Although you are done composing the two functions, it is necessary to simplify the answer if possible. Your Turn Problem #4

9. 9 1.6 Operations with Functions and Composition Solution: Your Turn Problem #5

10. 10 1.6 Operations with Functions and Composition Now,replace g(2) with -3. The following example will demonstrate how to evaluate a composition for a given value. Find f(-3). This will be the answer to (f  g)(2). Your Turn Problem #6 The End B.R. 1-6-07