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C3 Chapter 2 Functions

C3 Chapter 2 Functions. Dr J Frost (jfrost@tiffin.kingston.sch.uk). Last modified: 10 th September 2014. What is a mapping?. A mapping is something which maps an input value to an output value. Inputs. -1. -1. Outputs. 1. 0. f(x) = 2x + 1. 4.4. 1.7. 5. 2. 7.2. 3.1.

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C3 Chapter 2 Functions

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  1. C3 Chapter 2 Functions Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 10th September 2014

  2. What is a mapping? A mapping is something which maps an input value to an output value. Inputs -1 -1 Outputs 1 0 f(x) = 2x + 1 4.4 1.7 5 2 7.2 3.1 ... ... ! The domain is the set of possible inputs. ! The range is the set of possible outputs.

  3. What is a mapping? We can also illustrate a mapping graphically, where the axis is the input and the axis is the output . where

  4. What is a function? • A function is: a mapping such that every element of the domain is mapped to • exactly one element of the range. ? Notation: Function? y y For each value of (except 0), we get two values of ! For each input ( value), we only get one output ( value) x x  No  Yes No   Yes f(x) = 2x Domain: all real numbers f(x) = √x Domain: f(4) = 2 but f(4) = -2 also. This violates the definition of a function. No  Yes   No  Yes

  5. One-to-one vs Many-to-one While functions permit an input only to be mapped to one output, there’s nothing stopping multiple different inputs mapping to the same output. Type Description Example ? ? Many-to-one function Multiple inputs can map to the same output. f(x) = x2 e.g. f(2) = 4 f(-2) = 4 2 4 -2 ? ? Each output has one input and vice versa. One-to-one function f(x) = 2x + 1 5 2 7 3 9 4

  6. Example ? Sketch: is an element of Domain: ? …the set of real numbers We can use any real number as the input! Range: ? The output has to be positive, since it’s been squared. Type: ? Many-to-one BBro Tip: Note that the domain is in terms of and the range in terms of .

  7. Test Your Understanding ? Sketch: Domain: ? Presuming the output has to be a real number, we can’t input negative numbers into our function. Range: ? The output, again, can only be positive. ? Type: One-to-one

  8. Exercise Determine the domain, range and type of function/mapping, as well as a quick sketch of the graph. 3 2 1 ? ? ? 4 5 6 ? ? ? 8 9 7 ? ? ?

  9. Check Your Understanding So Far A function is: a mapping such that every element of the domain is mapped to exactly one element of the range. The domain is: the set of possible inputs. The range is: the set of possible outputs. Give an example of a one-to-one function: Any increasing or decreasing function, such as any linear function (e.g. ), exponential, . Also … Give an example of a cubic equation which is one-to-one: will do! Give an example of a cubic equation which is many-to-one: (think about its graph) What is the range of ? ? ? ? ? ? ? ?

  10. Composite Functions We can apply another function, say , to the output of . Bro Tip: Think of as . This will allow you to remember that we apply first and then .

  11. Examples Let , and . What is… ? ? ? ? ? ? ? ?

  12. Quickfire Examples Do in your head! 1 ? ? ? ? ? ? ? ? 2 ? ? ? ? 3 4 ? ? ? ? ? ? ? ? 5 ? ? ? ? 6

  13. The opposite: determining sequence of functions Sometimes you’re given a composite function, and have to determine what functions it is composed of. e.g. If , , Find in terms of the following functions: ? Bro Tip: We know and are going to come into it somehow. And given , it looks like we do first before . ? ?

  14. Exercise 2D Given that and , find expressions for the functions: If and , find the number(s) such that . The functions and are defined by , and . Find in terms of the functions: b) c) d) e) f) g) If and , prove that 1 ? ? ? ? ? 3 ? 5 ? ? ? ? ? ? ? 7 ?

  15. Inverse Functions The inverse of a function maps the output values back to the input values. Explain why the function must be one-to-one for an inverse function to exist: If the mapping was many-to-one, then the inverse mapping would be one-to-many. But this is not a function! ?

  16. Finding the Inverse Function If is defined as Find the range of . Calculate Sketch the graphs of both functions. State the domain and range of . What do you notice about the graphs of the two functions, and the domain/range of the two functions? ? c ? a Start with and make the subject, before swapping and . Domain: Range: The domain and range have swapped. Since and swaps to find the inverse, it’s equivalent to a reflection in the line . ? d b ? ? e

  17. Quickfire Inverses ? ? ? ? ? • A self-inverse function is a function that is the same as its inverse. • (Can you think of others?)

  18. Test Your Understanding Q Q Find Sketch and on the same axis. A function is defined as State a suitable domain. Find Find State the range of So ? ? Domain: So The domain of is the range of . So a b c

  19. Exercise 2E Q1a, c, d, e, Q2, Q3, Q4a, c, e, Q6, Q7

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