§ 1-2 Functions

1. § 1-2 Functions The student will learn about: functions, domain, range, linear functions, quadratic functions, functions used in a business setting, and some new calculator techniques.

2. Definition of a Function Def. – a function f is a rule that assigns to each number x in a set a number f (x). That is, for each element in the first set there corresponds one and only one element in the second set. That’s one – not two, not three, but one, 1, uno, un, eine, no more no less, but exactly one. Example – For the sale of x items there corresponds a revenue. The first set is called the domain, and the set of corresponding elements in the second set is called the range.

3. Functions as a Machine A function f may be thought of as a numerical procedure or “machine” that takes an “input” number x and produces an “output” number f(x), as shown below. The permissible input numbers form the domain, and the resulting output numbers form the range.

4. Functions Defined by Equations 2x – y = 1 x2 + y2 = 25 (Not a function – graph it.) Note if x = 3 then y = both 4 and – 4.

5. USING THE VERTICLE LINE TEST

6. Functions • The domain and range can be illustrated graphically.

7. Functions Defined by Equations Finding the domain: 1. Eliminate square roots of negatives This implies that x – 3 ≥ 0 or x ≥ 3. 2. Eliminate division by zero This implies that x – 5 ≠ 0 or x ≠ 5.

8. Introduction to Linear Functions A straight line has an equation of the form Ax + By = C.

9. Linear Functions And Equations.

10. Cost Function Finding a company’s cost function Cost Function C (x) = mx + b C is the total cost and mx is the variable cost. b (the y-intercept) is the fixed cost. m (slope) is the unit cost.

11. Example A company manufactures bicycles at a cost of $55 each. If the fixed cost are $900, express the company’s cost as a linear function. Cost Function C (x) = mx + b Where m (slope) is the unit costand b (the y-intercept) is the fixed cost. The unit cost is $55 and the fixed cost is $900 hence C (x) = 55 x + 900

12. Example The graph of C (x) = 55 x + 900 is a line with slope 55 and y-intercept 900, as shown below. The graph of C (x) = 55 x + 900 is a line with slope 55 and y-intercept 900, as shown below. The slope is the unit cost which is the same as the rate of change of the cost, which is also the company’s marginal cost. The graph of C (x) = 55 x + 900 is a line with slope 55 and y-intercept 900, as shown below. The slope is the unit cost which is the same as the rate of change of the cost, which is also the company’s marginal cost.The y-intercept is the fixed cost. Cost slope 55 C (x) = 55 x + 900 y-intercept 900 units

13. Introduction to Quadratic Functions A quadratic is an equation of the form f (x) = a x2 + b x + c , and graphs as a parabola.

14. Definition NOTE: a is the coefficient of the x 2 term, b is the coefficient of the x term, and c is the constant term.

15. Functions

16. Fact of Interest A quadratic function graph can either open up or open down. If a > 0 the graph opens up and if a < 0 the graph opens down. f (x) = - 2x 2 + 4x + 6 a = - 2 f (x) = x2 + x – 2. a = 1

17. Functions The vertex of a parabola is its “central” point, the lowest point on the parabola if it opens up and the highest point if it opens down.

18. Definition The vertex (lowest or highest point) is of importance and the x value can be found as follows: BUT - there is an easier way!

19. Graphing a Quadratic Function: Calculator 1. Turn the calculator on and press the y = button. If something is there press clear. x-intercepts using CALC and zero giving 1 and -2 y-intercept using CALC and value giving -2 2. Enter the function Y1 = x 2 + x – 2 and then press ZOOM and 6 for a standard window. 3. Find the x and y intercepts using the calculator. OR use table!

20. Graphing (Continued) Giving x = - 0.5 and y = - 2.25 4. The vertex of the parabola can be found on the calculator using the minimum or maximum menu options. 5. Press CALC and 3 for a minimum. y = x 2 + x – 2 NOTE: There is no need to use the previous formula for the x-value.      I love my calculator!       

21. Solving a Quadratic Equation f (x) = x2 + x – 2. 1. The solutions of a quadratic (the x-intercepts) are also called roots or zeros. We just found the x-intercepts using the calculator. 2. Solutions may also be found algebraically. Let f (x) = 0 and solve for x : a. Factor : 0 = (x + 2)(x – 1) and x = 1 and – 2. b. Complete the square – No thank you! c. Use the quadratic formula – Next slide please!

22. Solving a Quadratic Equation f (x) = x2 + x – 2. c. Use the quadratic formula : a = 1, b = 1, c = -2. OR use your calculator under calc and zeros.      I love my calculator!       

23. Quadratic Reviewy = ax 2 + bx + c x intercepts CALC & ZERO (x, 0) y intercept CALC & VALUE (0, y) opens up if a > 0 opens down if a < 0 vertex at CALC & MAX or MIN

24. Basic Business Applications. Cost FunctionC = Unit cost · Quantity + Fixed cost Where m (slope) is the unit costand b (the y-intercept) is the fixed cost. Revenue FunctionR = (price) · (quantity sold) (Note: quantity is number of items sold at price of $p.)

25. Basic Business Applications. Profit Function P = R – C Let C represent cost, R represent revenue and P profit. Then one of three things can occur: R > C P > 0 a profit Let C represent cost, R represent revenue and P profit. Then one of three things can occur: R > C P > 0 a profit R = C P = 0 a break-even point, or Let C represent cost, R represent revenue and P profit. Then one of three things can occur: R > C P > 0 a profit R = C P = 0 a break-even point, or R < C P < 0 a loss.

26. Application Example Break-even Analysis. Use the revenue and cost functions, R (x) = x (2,000 – 60x) & C (x) = 4,000 + 500 x, where x is thousands of dollars. Both functions have domain 0 ≤ x ≤ 25. 1. Sketch the graph of both functions on the same coordinate system.(Algebraically) Do you have any idea how long this would take or how difficult it is to do? • Find the break-even point (R (x) = C (x)) (Algebraically). Do you have any idea how long this would take or how difficult it is to do? Well do you?

27. 0 ≤ x ≤ 25 0 ≤ y ≤ 17,000 1. Graph both functions on your calculator. Application Example by Calculator Break-even Analysis. Use the revenue and cost functions, R (x) = x (2,000 – 60x) & C (x) = 4,000 + 500 x, where x is thousands of dollars. Both functions have domain 0 ≤ x ≤ 25. 2. Use Calc & intersect to find the break-even point. 3. When does profit occur? (3.03, 5518) (21.96, 14982)      I love my calculator!       

28. Solving Quadratic Equations The relationship among the cost, revenue, and profit functions can be seen graphically as follows. Notice that the break-even points correspond to a profit of zero, and that the maximum profit occurs halfway between the two break-even points.

29. Application Example by Calculator Break-even Analysis. This is the problem we just did! Use the revenue and cost functions, R (x) = x (2,000 – 60x) & C (x) = 4,000 + 500 x, where x is thousands of dollars. Both functions have domain 0 ≤ x ≤ 25. For homework, graph the profit equation with these two graphs – revenue and cost.      Embrace your calculator!      

30. Profi t Spreadsheet Exploration A spreadsheet may be substituted for a graphing calculator.

31. Summary. • We learned about functions and the basic terms involved with functions. • We learned about the linear functions. • We learned about the quadratic functions. • We learned about the basic business functions. • We learned how to use a graphing calculator to make our work much easier. • We saw how spreadsheets can replace the work of a graphing calculator.      I love my calculator!       

32. ASSIGNMENT §1.2 on my website