1. Concept of a Function

2. 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.

3. Consider the function y = x + 1.5 6
y = x + 1
Input x
Output y
y = 5 + 1

4. Consider the function y = x + 1.10 11
y = x + 1
Input x
Output y
y =
Independent variable
Dependent variable
The value of y depends on the value of x.

5. 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.

6. 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.

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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)

13. 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

14. 3. Graphical Representationx
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.

15. 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.

16. 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.

17. Notation of a Function

18. 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 =

19. 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

20. 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

21. 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.