1. 6.6 Function Operations Honors

2. Operations on Functions Addition: h(x) = f(x) + g(x) Subtraction: h(x) = f(x) – g(x) Multiplication: h(x) = f(x) g(x) Division: h(x) = f(x) / g(x)

3. Function Operations You can perform operations, such as addition, subtraction, multiplication, and division, with functions… For example: If f(x) = 3x and g(x) = x – 5 f(x) + g(x) = 3x + (x – 5) = 4x – 5 f(x) – g(x)= 3x – (x – 5) = 2x + 5 f(x) g(x) = 3x(x – 5) = 3x 2 – 15x What about f(x) ÷ g(x) ?

4. f(x) ÷ g(x) = 3x x – 5 Be sure to consider the domain (the possible inputs)… The domain does not include 5 since that would make the denominator 0… Therefore, the domain is all real numbers except x = 5. Example: Find the domain of Think of the values that will make the square root a negative number The domain is all real numbers greater than 1. The graph will confirm this...

5. Example f(x) = 3x + 2 g(x) = -4x – 1 Find the following along with their domains f(x) + g(x) = f(x) – g(x) =

6. Example f(x) = 3x + 2 g(x) = -4x – 1 Find the following along with their domains f(x) g(x) =

7. Examples 1.f(x) + g(x) = 2.f(x) – g(x) = Domains

8. Examples 1.f(x) g(x) = 2.f(x)/g(x) = Domains

9. f(x) = 3x g(x) = f(x) g(x) =

10. Example Let f(x) = x 2 – 9 and g(x) = x + 3, find f * g and, Find their domains

11. Composition of functions  Composition of functions is the successive application of the functions in a specific order.  Given two functions f and g, the composite function is defined by and is read “f of g of x.”  The domain of is the set of elements x in the domain of g such that g(x) is in the domain of f.  Another way to say that is to say that “the range of function g must be in the domain of function f.”

12. A different way to look at it… Function Machine x Function Machine g f

13. Example  Evaluate and when x = -3  f(x) = x – 5 and g(x) = x 2

14. Example  Evaluate and when x = 5  f(x) = x – 3 and g(x) = 2x 2 – 1

15. Exit Ticket Perform each operation listed and find their domain (#’s 1 – 3) 1.Let f(x) = 2x 2 + 8 and g(x) = x – 3 find f + g and f – g 2.Let f(x) = 2x + 5 and g(x) = x 2 – 3x + 2, find 3.Let f(x) = 7x + 5 and g(x) = x 2, find f(x) * g(x) 4. Let g(x) = 2x and f(x) = x 2 + 4. Find (f º g)(-2) and (g º f)(4)