Concept of a Function

Consider y = 2x. x y 1 2 2 4 3 6 … … Each value of x determines exactly one value of y. y is a function of x.

Consider the function y = x + 1. 5 6 y = x + 1 Input x Output y y = 5 + 1

Consider the function y = x + 1. 10 11 y = x + 1 Input x Output y y = 10 + 1 Independent variable Dependent variable The value of y depends on the value of x.

If y2 = 9x, is y a function of x? ∵ When x = 1, y = 3 or –3. i.e. There is more than one value of y for x = 1. ∴ y is not a function of x.

Each value of x gives exactly one value of y. Follow-up question Determine whether (where x ≠ 0) is a function of x. When x = 1, ; When x = 2, ; When x = 3, , etc. Each value of x gives exactly one value of y. ∴ y is a function of x.

Consider another function y = x2, where x = 1, 2, 3. Independent variable x 1 Collection of values that x can take is called the domain of the function. 2 3 domain

Consider another function y = x2, where x = 1, 2, 3. Independent variable Dependent variable x y Collection of values that y can take is called the range of the function. 1 1 2 4 3 9 domain range

Consider another function y = x2, where x = 1, 2, 3. Independent variable Dependent variable x y Collection of values that must include all possible values of y is called the co-domain of the function. 1 1 2 4 3 9 domain range co-domain

Follow-up question Consider the function y = 2x, where x = 1, 2, 3, 4, … Find its domain and range, and state one of its possible co-domains. Domain: collection of 1, 2, 3, 4, … Range: collection of values that y can take

Follow-up question Consider the function y = 2x, where x = 1, 2, 3, 4, … Find its domain and range, and state one of its possible co-domains. Domain: collection of 1, 2, 3, 4, … Range: collection of 2, 4, 6, 8, … Co-domain: all even numbers

Follow-up question Consider the function y = 2x, where x = 1, 2, 3, 4, … Find its domain and range, and state one of its possible co-domains. Domain: collection of 1, 2, 3, 4, … Range: collection of 2, 4, 6, 8, … Co-domain: 2, 4, 6, 8, … (or other possible answers like all positive numbers)

Different Representations of Functions In how many ways can we represent a function? 1. Algebraic Representation A table showing the relationship between x and y Express y in terms of x. e.g. y = 2x 2. Tabular Representation x 1 2 3 y 4 6

3. Graphical Representation x 1 2 3 6 4 2 y Plot the corresponding values of x and y on a coordinate plane. y = 2x From the graph, there is exactly one value of y for each value of x.

Follow-up question For each of the following graphs, determine whether y is a function of x, where 4  x  8 and y can be any real numbers. (a) y x ∵ Any vertical line intersects the graph at only one point. i.e. For any value of x where 4  x  8, there is only one corresponding value of y. 4 8 vertical lines ∴ y is a function of x.

Follow-up question For each of the following graphs, determine whether y is a function of x, where 4  x  8 and y can be any real numbers. (b) y x ∵ The vertical line on the left intersects the graph at two points. i.e. For a certain value of x where 4  x  8, there are more than one corresponding value of y. 4 8 vertical line ∴ y is not a function of x.

Notation of a Function

Apart from using y, we can use different notations to denote different functions. For example, y = 10 – x f(x) = 10 – x y = 2a – 1 g(a) = 2a – 1 3 2 t y = 3 ) ( 2 t H =

Find the values of the function when (a) x = 3, (b) x = –3. Consider f(x) = 10 – x. Find the values of the function when (a) x = 3, (b) x = –3. (a) When x = 3, f(3) = 10 – 3  When x = 3, the value of the function is f(3). = 7 (b) When x = –3, f(–3) =10 – (–3),  When x = –3, the value of the function is f(–3). = 13

Follow-up question If f(x) = ax + 8 and f(–2) = 2, find the value of a. ∵ f(–2) = 2 ∴ a(–2) + 8 = 2 –2a + 8 = 2 3 = a

In general, for a function y = f(x), (i) f(a) + f(b) ≠ f(a + b) (ii) f(a) – f(b) ≠ f(a – b) (iii) f(a)  f(b) ≠ f(ab) (iv) (v) kf(a) ≠ f(ka) where a, b and k are constants.