Chapter 1 - What is a Function.pdf
- 1. What is a Function Chapter 1 This Photo by Unknown Author is licensed under CC BY
- 2. A function 𝑓 is a rule that assigns to each element x in a set 𝐷 exactly one element, called 𝑓 𝑥 , in a set of 𝐸. • We usually consider functions for which the sets 𝐷 and 𝐸 are sets of real numbers. • The set 𝐷 is called the domain of the function. • The range of 𝑓(𝑥) is the set of all possible values of 𝑓(𝑥) as 𝑥 varies throughout the domain. • A symbol that represents an arbitrary number in the domain of a function 𝑓 is called an independent variable. • A symbol that represents a number in the range of 𝑓 is called a dependent variable. 2
- 3. THE VERTICAL LINE TEST A curve in the -plane is the graph of a function of if and only if no vertical line intersects the curve more than once. 3
- 4. Example 1.1 This Photo by Unknown Author is licensed under CC BY-SA-NC This Photo by Unknown Author is licensed under CC BY-SA Is it a Function?
- 5. Even and Odd Functions • If a function 𝑓 satisfies 𝑓 𝑥 = 𝑓(−𝑥) for every number in its domain, then 𝑓 is called an even function. • For example, 𝑓 𝑥 = 𝑥2 is even because 𝑓 −𝑥 = (−𝑥)2 = 𝑥2 = 𝑓(𝑥) • If 𝑓 satisfies 𝑓 −𝑥 = −𝑓(𝑥) for every number in its domain, then 𝑓 is called an odd function. • For example, 𝑓 𝑥 = 𝑥3 is odd because 𝑓 −𝑥 = (−𝑥)3 = −𝑥3 = −𝑓(𝑥) 5
- 6. Example 1.1 a) 𝑓 𝑥 = 𝑥5 + 𝑥 6 Determine whether each of the following functions is even, odd, or neither even nor odd. b) 𝑔 𝑥 = 1 − 𝑥4 c) ℎ 𝑥 = 2𝑥 − 𝑥2
- 7. Example 1.2.a 𝑓 −𝑥 = −𝑥5 + −𝑥 = (−1)5𝑥5 + −𝑥 = −𝑥5 − 𝑥 = − 𝑥5 + 𝑥 𝑓 −𝑥 = −𝑓 𝑥 Therefore, 𝑓 is an odd function. 7
- 8. Example 1.2.b 𝑔 −𝑥 = 1 − (−𝑥)4 = 1 − 𝑥4 𝑔(−𝑥) = 𝑔(𝑥) Therefore, 𝑔 is an even function. 8
- 9. Example 1.2.c ℎ −𝑥 = 2 −𝑥 − (−𝑥)2 = −2𝑥 − 𝑥2 𝑓 −𝑥 ≠ 𝑓 𝑥 𝑎𝑛𝑑 𝑓 −𝑥 ≠ −𝑓 𝑥 Therefore, ℎ is neither even or odd. 9 Notice that the graph of h is symmetric neither about the y-axis nor about the origin.
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- 11. Vertical Shift: y = 𝑓 𝑥 + 𝑘 • Shift the graph of 𝑓 up k units k > 0 • Shift the graph of 𝑓 down 𝑘 units k < 0 Horizontal Shift: y = 𝑓 𝑥 + ℎ Shift the graph of 𝑓 left h units k > 0 Shift the graph of 𝑓 right 𝑘 units k < 0 11
- 13. Stretching 𝑦 = 𝑐𝑓 𝑥 Stretch the graph vertically by c 𝑦 = 𝑓 𝑥 𝑐 Stretch the graph horizontally by c Compressing 𝑦 = 1 𝑐 𝑓 𝑥 Compress the graph vertically by c 𝑦 = 𝑓 𝑐𝑥 Compress the graph horizontally by c Reflection 𝑦 = −𝑓 𝑥 Reflects the graph across the x-axis 𝑦 = 𝑓 −𝑥 Reflects the graph across the y-axis 13
- 14. Assignment 1.1 1.1.1. 𝑠𝑙𝑜𝑝𝑒 = −6, 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 1,3 1.1.2.𝑠𝑙𝑜𝑝𝑒 = 3, 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ −3,2 1.1.3.𝑠𝑙𝑜𝑝𝑒 = 1 3 , 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 0,4 1.1.4.𝑠𝑙𝑜𝑝𝑒 = 2 5 , 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 8 1.1.5. 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 2,1 𝑎𝑛𝑑 −2, −1 1.1.6. 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ −3,7 𝑎𝑛𝑑 1,2 1.1.7. 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 5, 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = −3 1.1.8. 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = −6, 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 9 14 write the equation of the line satisfying the given conditions in slope-intercept form of the following expressions
- 15. Assignment 1.2 1.2.1. 𝑓 𝑥 = 2𝑥5 − 3𝑥2 + 2 1.2.2. 𝑓(𝑥) = 𝑥3 − 𝑥7 1.2.3. 𝑓(𝑥) = 𝑒−𝑥2 1.2.4. 𝑓(𝑥) = 1 + 𝑠𝑖𝑛𝑥 15 Determine whether 𝑓 is even, odd, or neither even nor odd.