2. MAT 150 Module 7 – Operations With Functions Lesson 1 – Combining Functions

3. Combining Functions Say we have two functions, f(x) and g(x). There are four ways we can combine the two functions: we can find their sum, difference, product, and quotient.

4. Combining Functions The sum of two function is called (f+g)(x). The difference of two function is called (f-g)(x). The product of two functions is called (fg)(x). The quotient of two functions is called (f/g)(x).

5. Combining Functions The quotient of two functions, (f/g)(x), may have domain issues because we can’t have the denominator equal to 0. We need to exclude any x values that make the denominator, g(x), equal to 0, from the domain.

6. Example 1 For the functions f(x) = 3x + 5 and g(x) = x 2 + 4x + 4, find: The sum, (f + g)(x) The difference, (f-g)(x) The product, (fg)(x) The quotient, (f/g)(x).

7. Example 1 – Solution For the functions f(x) = 3x + 5 and g(x) = x 2 + 4x + 4, find the sum, (f+g)(x). (f+g)(x) = 3x + 5 + x 2 + 4x + 4 (f+g)(x) = x 2 + 7x + 9

8. Example 1 – Solution For the functions f(x) = 3x + 5 and g(x) = x 2 + 4x + 4, find (f - g)(x). (f - g)(x) = 3x + 5 – (x 2 + 4x + 4) (f - g)(x) = 3x + 5 – x 2 - 4x - 4 (f - g)(x) = -x 2 - x + 1

9. Example 1 - Solution For the functions f(x) = 3x + 5 and g(x) = x 2 + 4x + 4, find (f ● g)(x). (f ● g)(x) = (3x + 5)(x 2 + 4x + 4). (f ● g)(x) = 3x 3 + 12x 2 + 12x + 5x 2 + 20x + 20 (f ● g)(x) = 3x 3 + 17x 2 + 32x + 20.

10. Example 1- Solution For the functions f(x) = 3x + 5 and g(x) = x 2 + 4x + 4, find (f/g)(x). That’s it! There’s nothing we can do to simplify.

11. Example 1 Continued What is the domain of ?

12. Example 1 Continued What is the domain of ? Set g(x) = 0: x 2 + 4x + 4 = 0. Factor: (x + 2)(x + 2) = 0. x+2 = 0 has solution: x = -2 The domain of (f/g)(x) is all real numbers except x = -2.

13. Applications of Cost Functions In the reading, we learned about two applications of combining functions: The profit function and the average cost function. Let’s do an example to practice these concepts.

14. Applications of Combined Functions - Example A company’s cost for producing x items is given by the function C(x) = 15x + 5000 and their revenue for selling x items is given by R(x) = 80x. a) Find the profit function, P(x). b) How much profit would be earned by selling 100 items? c) How many items must the company sell to break even?

15. Applications of Combined Functions - Solution A company’s cost for producing x items is given by the function C(x) = 15x + 5000 and their revenue for selling x items is given by R(x) = 80x. a)Find the profit function, P(x). Recall that P(X) = R(x) – C(X). So P(X) = 80x – (15x + 5000) P(X) = 80x – 15x – 5000 P(X) = 65x – 5000.

16. Applications of Combined Functions - Solution b) How much profit would be earned by selling 100 items? To find the profit earned by selling 100 items, find P(100). P(x) = 65x – 5000 P(100) = 65(100) – 5000 P(100) = 6500 – 5000 = 1500. The profit earned by selling 100 items is $1500.

17. Applications of Combined Functions - Solution c) How many items must the company sell to break even? The company will break even when profit is $0, so we want P(X) = 0. To find the number of items the company must sell to break even, set P(X) = 0 and solve for x. 65x – 5000 = 0 65x = 5000 x = 76.923 ≈ 77. The company must sell 77 items to break even.

18. Applications of Combined Functions - Solution

19. Applications of Combined Functions - Example A company’s cost for producing x items is given by the function C(x) = 15x + 5000 and their revenue for selling x items is given by R(x) = 80x. a) Find the average cost function b) What is the average cost if the company produced 100 items? c) What is the domain of the average cost function? d) Graph the average cost function on the window [0, 10], x scl = 1 by [0, 5000], y scl = 500.

20. Applications of Combined Functions - Solution a)Find the average cost function, Recall that the average cost function is That is, we take the cost function C(x) and divide it by the number of items, x. If C(x) =15x + 5000, then That’s it!

21. Applications of Combined Functions - Solution b) What is the average cost if the company produced 100 items?

22. Applications of Combined Functions - Solution c) What is the domain of the average cost function? Remember we are looking for the set of x values that we are allowed to input into the function. Since x represents the number of items produced, we must exclude negative x values because we cannot produce negative items. We also must exclude x = 0 since this substituting x = 0 would cause us to divide by 0. So the domain is all positive real numbers not including 0, which we would write as (0, ∞).

23. Applications of Combined Functions - Solution