1. Functions A function is a relationship between two sets: the domain (input) and the range (output). 1234512345 4 7 10 13 16 DomainRange Input Output This function can be expressed in function notation as or

2. The graph of y = f(x)= 3x + 1 x f(x) Domain and Range The domain: The set of inputs (x) for a function is called the domain of the function. The range: The set of output numbers (y) is called the range of the function. or Since for each input number (x), there is only one output number (y) this relationship is called one - to - one relationship. x belongs to the set of real numbers

3. Many – to – one relationships Many values of input ( x ) map to one value of output (y). f(x) x A horizontal line cuts the graph more than once. A vertical line cuts the graph at most once The function is y = f(x) = - x (x + 1)(x – 2)

4. One – to – many relationships One value of input ( x ) maps to more than one values of output (y). y =f(x) x A horizontal line cuts the graph at most once. A vertical line cuts the graph more than once. y is not a function of x. To be a function, a relationship must be one-one or many – one and it must be defined for all values of the domain.

5. Examples Which of the following are functions? (i) (ii) (i) f(x) = 2x + 3 is a function (one to one) and its range is f(x)  . Solution is not a function because it undefined when x = 1. (ii) x f(x) f(x) is undefined at this point.

6. Restricting the domain of a function Example Now it is a function because 1 is excluded from the domain and its range is f(x)  ., f(x)  0 Find the largest possible domain of each of the following functions. Largest possible domains Example (i) x  0 (ii) x  3(iii) x  ., x  3 (iv) x > 1 (v) x  ., x  2, x  -3

7. The range Example Find the range of the following functions. Solution (i) f(x)  -2(ii) f(x)  5iii) -2  f(x)  2 f(x)  -11 Completing the square Example Find the range of the following function. x 2 – 4x + 6 = (x – 2) 2 – 4 + 6 = (x – 2) 2 + 2  f(x)  2

8. The range for a given domain Example Find the range of f(x) = sin x for 0   x  60  y =f(x) x Domain Range