2.Functions02.pdf
- 1. • Be able to understand: • What is a constant function? • What is an Identity function? • What is polynomial function? • What is an absolute function? • What is square root function? • What is piecewise function? • What is greatest integer function? • What is least integer function? • What is increasing and decreasing function? Dr Sumaira Rehman
- 10. Draw the graph of the function y=𝟏 + 𝒙𝟐 And write the domain and range ? Dr Sumaira Rehman
- 13. Small Exercise about shifting the absolute function
- 15. Write the domain and range of the function y= 𝑥 − 3 Write the domain and range of the function y=1- 𝒙
- 17. A piecewise function is a function where more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all of these smaller domains. Piecewise function:
- 18. Example 1:
- 20. Example 2:
- 26. Least integer function (Ceiling function) Also known as the ceiling function, LIF(x) is the least integer function, which returns the value of the least integer more than or equal to x. For example, LIF(3.55) will return a value 4. For example ⌈3.578⌉ = 4 ⌈0.78⌉ = 1 ⌈-4.64⌉ = - 4
- 40. 3. Composite function : When the output from one function is used as the input to another function we form what is known as a composite function.
- 41. FIGURE 3.1 : Arrow diagram for ƒ o g.
- 42. Example 3.1
- 44. Here the function rule is 'multiply the input by 3 and then add 2'. The composite function f (f (x)) is illustrated in Figure 3.2. Example 3.2 Solution Figure 3.2 The composition of f (x) with itself.
- 47. Shifting the graph of the function:
- 51. To shift the graph ƒ(x)=𝒙𝟐 of up (or down), we add positive (or negative) constants to the formula for ƒ
- 52. To shift the graph of to the y=𝒙𝟐 left, we add a positive constant to x. To shift the graph to the right, we add a negative constant to x
- 53. Adding -2 to x in y= 𝐱 and then -1 adding to the result, gives y= 𝐱 − 𝟐 − 𝟏 and shifts the graph 2 units to the right and 1 unit down Shifting the graph of the function y= 𝐱
- 64. The trigonometric functions are important because they are periodic, or repeating, and therefore model many naturally occurring periodic processes.
- 82. Exponential functions have the form f(x) = bx, where b > 0 and b ≠ 1. Just as in any exponential expression, b is called the base and x is called the exponent.
- 83. An example of an exponential function is the growth of bacteria. Some bacteria double every hour. If you start with 1 bacterium and it doubles every hour, you will have 2x bacteria after x hours. This can be written as f(x) = 2x. Example With the definition f(x) = bx and the restrictions that b > 0 and that b ≠ 1, the domain of an exponential function is the set of all real numbers. The range is the set of all positive real numbers. The following graph shows f(x) = 2x.
- 86. Logarithmic Functions These are the functions 𝐟 𝐱 = 𝐥𝐨𝐠𝐚 𝐱 where the base 𝐚 ≠ 𝟏 is a positive constant. They are the inverse functions of the exponential functions.