The height of a ball thrown vertically upwards with initial speed 20ms -1 after time t is h = 20t – 5t 2. The formula gives h in terms of t, and assigns to each value of t between 0 and 4 (why?) a unique value of h between… (what?) The idea of a function FUNCTIONS

The height of a ball thrown vertically upwards with initial speed 20ms -1 after time t is h = 20t – 5t 2. The formula gives h in terms of t, and assigns to each value of t between 0 and 4 (why?) a unique value of h between… (what?) Solution Why? When h = 0, the ball must be at the point from which it was thrown. So,0 = 20t – 5t 2 0 = 5t(4 – t)→t = 0 or t = 4. What? Maximum height occurs when = 0. = 20 – 10t 20 – 10t = 0 when t = 2 So, when t = 2, h = 20(2) – 5(2) 2 = 20m. So, h ranges from 0m to 20m. The idea of a function FUNCTIONS dh dt dh dt

This is an example of a function: notation h(t). Functions are examples of a wider class called mappings. The idea of a function FUNCTIONS

For example, Why fly to Geneva in January? Several people arriving at Geneva airport from London were asked the main purpose of their visit. Their answers were recorded. David JoanneSkiing JonathanReturning home LouiseTo study abroad PaulBusiness Shamaila Karen The language of functions FUNCTIONS

This is an example of a mapping. A mapping is any rule which associates two sets of items. In this example each of the names on the left is an object or input, and each of the reasons on the right is an image, or output. The language of functions FUNCTIONS

For a mapping to make sense or to have any practical application, the inputs and outputs must each form a natural collection or set. The set of possible inputs (in this case, all the people who flew to Geneva from London in January) is called the domain of the mapping. The set of possible outputs (in this case, the set of all possible reasons for flying to Geneva) is called the co-domain of the mapping. The language of functions FUNCTIONS

The seven people questioned in this example gave a set of four reasons, or outputs. These form the range of the mapping for this particular set of inputs. The range of any mapping forms part or all of its co-domain. The language of functions FUNCTIONS

Notice that Jonathan, Louise and Karen are all visiting Geneva on business: each person only gave one reason for the trip, but the same reason was given by several people. This mapping is said to be many-to-one. A mapping can also be one-to-one, one-to-many or many-to-many. The language of functions FUNCTIONS

The relationship between the people and their UK passport numbers will be one-to- one. The relationship between the people and their items of luggage is likely to be one-to- many, and that between the people and the countries they have visited in the last 10 years will be many-to-many. The language of functions FUNCTIONS

In Mathematics, many (but not all) mappings can be expressed using algebra. Here are some examples of Mathematical mappings. For each of these examples: Decide whether the mapping is one-to-one, many-to-many, one-to-many or many-to -one. Mappings FUNCTIONS

Sets of numbers Real numbers ℝ, ℝ +, ℝ - Rational Numbers ℚ, ℚ +, ℚ - Integers ℤ, ℤ +, ℤ - Natural Numbers ℕ FUNCTIONS Sets of numbers ( ℝ ) Real numbers Irrational numbers: √2, π, e. ( ℚ ) Rational numbers ⅓, -3, 1.4, 0.25 ( ℤ ) Integers -3, -2, -1, 0, 1, 2, 3 ( ℕ ) Natural numbers 0,1,2,3,4…

Domain: integersCo-domain: real numbers ObjectsImages General rule: x2x + 5 Mappings Example 1

Domain: integersCo-domain: real numbers ObjectsImages General rule: x2x + 5 Mappings Example 1 One-to-one

Domain: integersCo-domain: real numbers ObjectsImages π General rule: RoundedUnrounded whole numbersnumbers Mappings Example 2

Domain: integersCo-domain: real numbers ObjectsImages π General rule: RoundedUnrounded whole numbersnumbers Mappings Example 2 One-to-many

Domain: real numbersCo-domain: real numbers ObjectsImages General rule: x°sin x° Mappings Example 3

Domain: real numbersCo-domain: real numbers ObjectsImages General rule: x°sin x° Mappings Example 3 many-to-one

Domain: quadraticCo-domain: real numbers equations with real roots ObjectsImages x 2 – 4x + 3 = 00 x 2 – x = 01 x 2 – 3x + 2 = General rule: ax 2 + bx + c = 0 x = x = Mappings Example 4 -b - √(b 2 – 4ac) 2a -b - √(b 2 – 4ac) 2a

Domain: quadraticCo-domain: real numbers equations with real roots ObjectsImages x 2 – 4x + 3 = 00 x 2 – x = 01 x 2 – 3x + 2 = General rule: ax 2 + bx + c = 0 x = x = For practice go to: password: clvmaths expert – functions 1 – mapping diagrams Mappings Example 4 -b - √(b 2 – 4ac) 2a -b - √(b 2 – 4ac) 2a many-to-many

Mappings which are one-to-one or many-to-one are of particular importance, since in these cases there is only one possible image for every object. Mappings of these types are called functions. For example, x → x 2 and x → cos x° are both functions, because in each case for any value of x there is only one possible answer. The mapping of rounded whole numbers onto un-rounded numbers is not a function, since, for example, the rounded number 5 could be the object for any un-rounded number between 4.5 and 5.5. There are several different equivalent ways of writing a function. For example, the function which maps x onto x 2 can be written in any of the following ways. y = x 2 f(x) = x 2 f:x → x 2 Functions FUNCTIONS Read this as ‘f maps x onto x 2 ‘

It is often represent a function graphically, as in the following examples, which also illustrate the importance of knowing the domain. Functions FUNCTIONS

Functions Example 5 Sketch the graph of y = 3x + 2 when the domain of x is: a) x  ℝ b) x  ℝ + (i.e. positive real numbers) c) x  ℕ

Functions Answers a) x  ℝ b) x  ℝ + c) x  ℕ When the domain is ℝ, all values of y are possible. The range is therefore ℝ, also. When x is restricted to positive values, all the values of y are greater than 2, so the range is y > 2. In this case the range is the set of points {2, 5, 8, …}. These are clearly all of the form 3x + 2 where x is a natural number (0, 1, 2, …). This set can be written neatly as {3x + 2: x  ℕ }. The open circle shows that (0, 2) is not part of the line y x y x y x

Functions Example 6 Sketch the graph of the function y = f(x) f(x) = x 2 when the domain of x is 0 ≤ x < 3

Functions Answer Sketch f(x) = x 2 0 ≤ x < 3 y x The closed circle shows that (0, 0) is part of the line Restricting the domain has produced a one-to-one function here. For practice go to: password: clvmaths expert – functions 2 – domain

Functions FUNCTIONS When you draw the graph of a mapping, the x co-ordinate of each point is an input value, the y coordinate is the corresponding output value. The table below shows this for the mapping x → x 2, or y = x 2, and the figure shows the resulting points on a graph. Input (x)Output (y)Point plotted -24(-2, 4) 1(-1, 1) 00(0, 0) 11(1, 1) 24(2, 4) y x

If a mapping is a function, there is one and only one value of y for every value of x in the domain. Consequently, the graph of a function is a simple curve or line going from left to right, with no doubling back. Functions FUNCTIONS y = x 2 y = ± √ 9 – x 2 

Is it a function? Question 1 Does this graph show a function? Yes / No x y

Is it a function? Question 2 Does this graph show a function? Yes / No x y

Is it a function? Question 3 Does this graph show a function? Yes / No x y

Is it a function? Question 4 Does this graph show a function? Yes / No x y

Is it a function? Question 5 Does this graph show a function? Yes / No x y

Is it a function? Question 6 Does this graph show a function? Yes / No x y

Is it a function? Question 7 Does this graph show a function? Yes / No x y

Is it a function? Question 8 Does this graph show a function? Yes / No x y

Is it a function? Question 9 Does this graph show a function? Yes / No x y

Is it a function? Question 10 Does this graph show a function? Yes / No x y

Is it a function? Answers 1. No (a one to many mapping) 2. Yes (a one to one mapping) 3. No (a one to many mapping) 4. Yes (a one to one mapping) 5. Yes (a one to one mapping) 6. No (a many to many mapping) 7. Yes (a one to one mapping) 8. Yes (a many to one mapping) 9. Yes (a one to one mapping) 10. Yes (a many to one mapping)

Functions Mini - Review What is the domain of a function? FUNCTIONS A rule that is a function… A rule that is not a function… Match each letter to the correct set of numbers. Natural numbers Real numbers Rational numbers Integers ℝℚℤℕℝℚℤℕ Show two different ways to write the same function…