2. 2.4 – Operations with Functions  Objectives: Perform operations with functions to write new functions Find the composition of two functions  Standard: 2.8.11.S. Analyze properties and relationships of functions

3. I. Operations With Functions  For all functions f and g:  Sum (f + g)(x) = f(x) + g(x)  Difference (f – g)(x) = f(x) – g(x)  Product (f · g)(x) = f(x) · g(x)  Quotient ( )(x) =, where g(x) ≠ 0

4. Example 1

5. Example 2

6. Example 3

7. Example 4

8. Composition of Functions  Let f and g be functions of x.  The composition of f and g, denoted f ◦ g, is defined by f(g(x)).  The domain of y = f(g(x)) is the set of domain values of g whose range values are the domain of f. The function f ◦ g is called the composite function of f with g.

9. Example 1

10. Example 2 Let f(x) = -2x + 3 and g(x) = -2x. a. Find f ◦ g. b. Find g ◦ f. Let f(x) = x 2 + 4 and g(x) = 2x. a. Find f ◦ g. b. Find g ◦ f. Example 3 2 b.g(f(x)) = -2 (-2x +3) = 4x - 6 2 2 a. f(g(x)) = -2 (-2x) + 3 = -8x + 3 2 2

11. Example 4

12. Example 5  A local computer store is offering a $40.00 rebate along with a 20% discount. Let x represent the original price of an item in the store. a. Write the function D that represents the sale price after a 20% discount and the function R that represents the sale price after the $40 rebate. b. Find the composition functions (R ° D)(x) and (D ° R)(x), and explain what they represent. a.Since the 20% discount on the original price is the same as paying 80% of the original price, D(x) = 0.8x The rebate function is R(x) = x - 40 b.20% discount first$40 rebate first R(D(x)) = R(0.8x)D(R(x)) = D(x – 40) = (0.8x) – 40 = 0.8(x – 40) = 0.8x – 40 = 0.8x – 32 Notice that taking the 20% discount first results in a lower sales price.

13. End Section 2.4

14. Writing Activities  5. What is the difference between (fg)(x) and (f ◦ g)(x)? Include examples to illustrate your discussion.  6. In general, are (f ◦ g)(x) and (g ◦ f)(x) equivalent functions? Explain.

16. Homework Integrated Algebra II- Section 2.4 Level A Academic Algebra II- Section 2.4 Level B