Functionsbsitact
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The document discusses functions and their use in business contexts. It defines a function as a rule that pairs each element in one set to a single element in another set. Functions can be represented by ordered pairs in a table or graphically. Real-life examples of functions include the price of grocery items or a company's monthly sales. The independent variable is the input to a function, while the dependent variable is the output. Functions in business are often described using equations like cost, profit, or accumulated value functions. The document also discusses different types of functions like linear, quadratic, exponential, logarithmic, and polynomial functions.
Functionsbsitact
- 1. C A L C UL US 4 B S I T - A C T FUNCTIONS
- 2. CALCULUS FUNCTIONS Functions – is a rule of correspondence between two sets X and Y such that each element in the first set X is paired to one and only element in the second set Y X Y x y f Figure 1.1 Function as a Rule A B BA to
- 3. Key words: pairing and only one element y in Y is paired with each x in X. a b m N o Not a function a b m N o a function CALCULUS FUNCTIONS
- 4. Functions produce a set of ordered pairs(x,y) . 2 3 5 4 9 25 f F={(2,4), (3,9), (5, 25)} CALCULUS FUNCTIONS
- 5. Fundamental Theorem of Analytic Geometry Each ordered pair of real numbers correspond to a point (x,y) on the Cartesian plane. Graph – the collection of points on the Cartesian plane (graph of the function CALCULUS FUNCTIONS
- 6. • F={(2,4), (3,9), (5, 25)} CALCULUS FUNCTIONS
- 7. Functions in real life situations A. A student earns a general-weighted average upon graduation in LPU. B. Each item in a grocery has a price. C. Each government employee has corresponding asset and liability as stated in the government SALN. CALCULUS FUNCTIONS
- 8. Functions in real life (x,y) 𝑦 = 𝑓 𝑥 Domain – is the set of all admissible values of x. Range – is the set of all resulting values of y CALCULUS FUNCTIONS
- 9. Identify the domain and range for the following functions. A. A student earns a general-weighted average upon graduation in LPU. B. Each item in a grocery has a price. C. Each government employee has corresponding asset and liability as stated in the government SALN. D. Monthly sales report of a Camella Homes CALCULUS FUNCTIONS
- 10. Given a function 𝑦 = 𝑓 𝑥 Independent variable – X Dependent variable – Y Identify the dependent and independent variable in the following A. A student earns a general-weighted average upon graduation in LPU. B. Each item in a grocery has a price. C. Each government employee has corresponding asset and liability as stated in the government SALN. D. Monthly sales report of a Camella Homes CALCULUS FUNCTIONS
- 11. More on graphs. Not all graphs of functions in real life are continuous curves, especially in business. Monthly sales in hundred thousands Months (1-Jan, 2-Feb,. . .et. CALCULUS FUNCTIONS
- 12. Why would distinct points be connected by a curve? a. To show trending. b. To effectively show the behavior of the function c. To fill-in missing data with educated guesses. CALCULUS FUNCTIONS
- 13. Functions defined by equations In mathematics and business related situations, functions are described using equations. 𝑓 𝑥 = 𝐶𝑥 + 𝐷 where C is the cost per unit of production and D is the fixed cost. (Cost function) 𝑔 𝑡 = 𝐼(1.08) 𝑡 where I is the initial investment and t, number of years. (Accumulated value function) CALCULUS FUNCTIONS
- 14. 𝑅 𝑥 = 𝑝𝑥 − 𝑐𝑥 = 𝑝 − 𝑐 𝑥 Profit function, p is the price per unit of merchandize sold, and c is the cost per unit. CALCULUS FUNCTIONS
- 15. Input-the value of the independent variable Output – the value of the dependent variable CALCULUS FUNCTIONS
- 16. CALCULUS FOR BUSINESS FUNCTION Types of functions a. Linear function 𝑦 = 𝑎𝑥 + 𝑏
- 17. b. Quadratic function 𝑦 = 𝑎𝑥2 − 𝑏𝑥 + 𝑐 CALCULUS FUNCTIONS
- 18. c. Exponential function 𝑦 = 𝐶𝑏 𝑥, 𝐶 initial value, 𝑥 ∈ 𝑅 𝑦 = 2𝑒2 CALCULUS FUNCTIONS
- 20. CALCULUS FOR BUSINESS FUNCTIONS c. Log function (Logistic function) 𝑦 = 𝑙𝑜𝑔 𝑏 𝑥, 𝑏 ∈ 𝑅+, 𝑥 ∈ 𝑅 𝑦 = ln 𝑥, base e 𝑦 = log 𝑥, base 10
- 21. 𝑦 = ln 𝑥 CALCULUS FUNCTIONS
- 22. d. Polynomial function 𝑦 = 𝑎𝑥 𝑛 + 𝑏𝑥 𝑛−1 + ⋯ 𝑐 𝑥 ∈ 𝑅, 𝑎, 𝑏, … 𝑐, ∈ 𝑅 coefficients 𝑛 ∈ 𝑁 CALCULUS FUNCTIONS
- 23. • 𝑦 = 5 − 𝑥 + 2𝑥2 − 4𝑥3 CALCULUS FUNCTIONS
- 24. CALCULUS FUNCTIONS Conditional function (piece-wise functions) condition3if 2conditionif 1conditionif xh xg xf y