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.

22. Some Common Functions and their Graphs – Constant Functions & Linear Functions

23. What is a constant function?
Constant Functions
What is a constant function?
A function in the form y = c or f(x) = c, where c is a constant, is called a constant function.

24. Constant Functions What is a constant function?
For example, y = 1 and f(x) = –2 are constant functions.

25. Consider the constant function y = 1. x –2 2 y 12 y 1
◄ The value of y is always equal to 1, for any value of x. y x
(–2, ) 1
(0, ) 1
(2, ) 1
y = 1
y-coordinates of all the points on the line are equal to 1
A straight line parallel to the x-axis

26. Graph of y = c O y x
c 0 O y x
c= 0 O y x
c 0
(0, c)
(0, c)
(0, c)
The graph of a constant function is a horizontal line passing through (0, c).

27. Follow-up question
The figure shows the graph of a constant function. Write down the algebraic representation of the function. x y
2  2 4
∵ The graph is a horizontal line which cuts the y-axis at (0, –3).
(0, –3)
∴ The required function is y = –3.
y = f(x)

28. Linear Functions
A function in the form y = ax + b or f(x) = ax + b, where a and b are constants and a  0, is called a linear function of x.
Example:
(i) y = 3x – 1
linear functions of x
(ii) f(x) = –2x + 5

29. The graph cuts the x-axis at P.
Graph of y = ax + b O y x
y = ax + b
The graph cuts the x-axis at P. P

30. The graph cuts the y-axis at Q.
Graph of y = ax + b O y x
y = ax + b Q
The graph cuts the y-axis at Q.
x-intercept a b -

31. Graph of y = ax + b x-intercept y-intercept
The graph of a linear function is a straight line. It has only one x-intercept and one y-intercept.

32. Graph of y = ax + b a 0 a 0 O y x O y x
The value of y increases as x increases.
The value of y decreases as x increases.

33. Follow-up question
The figure shows the graph of the linear function y = f(x). Write down the x-intercept and the y-intercept of the graph. x y
2  4 2 2 4
∵ The graph cuts the x-axis and the y-axis at (2, 0) and (0, 3) respectively.
y = f(x)
(0, 3)
(2, 0)
∴ x-intercept = 2 and
y-intercept = 3

34. Some Common Functions and their Graphs – Quadratic Functions

35. quadratic functions of x
A function in the form y = ax2 + bx + c or f(x) = ax2 + bx + c, where a, b and c are constants and a  0, is called a quadratic function of x.
Example:
(i) y = –2x2 + x
quadratic functions of x
(ii) f(x) = (x + 3)2 + 5

36. Graph of y = ax2 + bx + c (a > 0)
axis of symmetry
The graph has reflectional symmetry about y x
opens upwards for a > 0
a vertical line c
The graph and its axis of symmetry intersect at .
y-intercept = c
a point
x-intercepts
vertex
(It is the minimum point of the graph.)
minimum value of y

37. Graph of y = ax2 + bx + c (a < 0)
axis of symmetry y x
vertex
(It is the maximum point of the graph.)
maximum value of y
opens downwards for a < 0
x-intercepts c
y-intercept = c

38. Coordinates of the vertex = (–1, 0)
Can you find the axis of symmetry and the coordinates of the vertex of the graph of y = x2 + 2x + 1?
axis of symmetry x y
4 3 2 1 1 2 8 6 4 2
y = x2 + 2x + 1
Axis of symmetry:
x = –1
vertex
Coordinates of the vertex = (–1, 0)

39. Follow-up question
Consider the graph of y = (x + 1)(5 – 2x). (a) Determine the direction of opening. (b) Find the x-intercept(s) and the y-intercept of the graph.
y = (x + 1)(5 – 2x)
(a)
y = 5x + 5 – 2x2 – 2x
∴ y = –2x2 + 3x + 5
∵ Coefficient of x2 = –2  0
∴ The graph opens downwards.

40. + (b) ∵ y = –2x2 + 3x + 5 ∴ The y-intercept of the graph is 5.
The x-intercepts of the graph are the roots of (x + 1)(5 – 2x) = 0. ) 2 5 )( 1 ( = - + x 2 5 or 1 = - + x 2 5 or 1 = - x
∴ The x-intercepts of the graph are –1 and . 2 5

41. Quadratic Functions in the Form y = a(x – h)2 + k

42. Can we obtain the features of the graph of a quadratic function without plotting its graph?
Yes. For the function y = a(x – h)2 + k, we can obtain the axis of symmetry and the coordinates of the vertex directly.

43. Let us study quadratic functions in the form y = a(x – h)2 + k.
Can we obtain the features of the graph of a quadratic function without plotting its graph?
Let us study quadratic functions in the form y = a(x – h)2 + k.

44. y = a(x – h)2 + k Case 1 : a > 0 a > 0 and (x – h)2  0O y x
a > 0 and (x – h)2  0
a(x – h)2  0
a(x – h)2 + k  k

45. The minimum value of y is k when x = h.
y = a(x – h)2 + k
axis of symmetry: x = h
Case 1 : a > 0 O y x
a > 0 and (x – h)2  0
a(x – h)2  0
y  k
minimum
value of y
The minimum value of y is k when x = h. k
vertex: (h, k)

46. y = a(x – h)2 + k Case 2 : a < 0 a < 0 and (x – h)2  0O y x
a < 0 and (x – h)2  0
a(x – h)2  0
a(x – h)2 + k  k

47. The maximum value of y is k when x = h.
y = a(x – h)2 + k
vertex: (h, k)
Case 2 : a < 0 O y x
a < 0 and (x – h)2  0
maximum
value of y k
a(x – h)2  0
y  k
The maximum value of y is k when x = h.
axis of symmetry: x = h

48. Maximum or minimum value
Features of the function y = a(x – h)2 + k and its graph
a > 0
a < 0
Direction of opening
Axis of symmetry
Vertex
Maximum or minimum value
upwards
downwards
x = h
(h, k)
(min. point)
(max. point)
min. value = k
max. value = k

49. ∵ Coefficient of x2 = 1 > 0
Can you find the minimum value of the function y = (x + 5)2 + 7 and the axis of symmetry of its graph?
y = a(x – h)2 + k
y = (x + 5)2 + 7
= [x – (–5)]2 + 7
◄ a = 1, h = –5, k = 7
∵ Coefficient of x2 = 1 > 0
min. value = k
∴ The minimum value of y is 7.
∴ The axis of symmetry is x = –5.
x = h

50. Follow-up question
For the function y = 9 – 2(x + 1)2, find (a) its maximum or minimum value, (b) the coordinates of the vertex.
(a) y = 9 – 2(x + 1)2
= –2(x + 1)2 + 9
For y = a(x – h)2 + k, coordinates of the vertex: (h, k)
= –2[x – (–1)]2 + 9
∵ Coefficient of x2 = –2 < 0
∴ The maximum value of y is 9.
(b) Coordinates of the vertex = (–1, 9)

51. Using the Algebraic Method to Find Optimum Values of Quadratic Functions

52. Forming Perfect Squares
Consider x2 + 6x.
By adding , we have
x2 + 6x
= x2 + 2(3)x + 32
◄ x2 + 2mx + m2 ≡ (x + m)2
= (x + 3)2
a perfect square
This process is called completing the square.

53. Completing the square To complete the squares for x2 + kx and x2 – kx,
add to each expression. Then

54. How to find the optimum value of the quadratic function y = ax2 + bx + c?
Convert y = ax2 + bx + c to the form y = a(x – h)2 + k by completing the square first.

55. Optimum values of quadratic functions
y = ax2 + bx + c
y = a(x – h)2 + k
completing
the square b 2a +  2

56. Optimum values of quadratic functions
y = ax2 + bx + c
y = a(x – h)2 + k
completing
the square
If a > 0, then the minimum value of y is k when x = h.
If a < 0, then the maximum value of y is k when x = h.

57. Can you find the optimum value of the function y = x2 + 2x + 2?
Complete the square for x2 + 2x. 2 ø ö ç è æ
Add , i.e. 12.
= x2 + 2x + 12 –
= (x2 + 2x + 1) – 1 + 2
= (x + 1)2 + 1
∵ Coefficient of x2 = 1 > 0
∴ The minimum value of y = x2 + 2x + 2 is 1.

58. Follow-up question
Find the optimum value of the quadratic function y = –4x2 + 24x – 15, and the corresponding value of x.
y = –4x2 + 24x – 15
= –4(x2 – 6x) – 15
Complete the square for x2 – 6x. 2 6 ø ö ç è æ
Add , i.e. 32.
= –4(x2 – 6x + 32 – 32) – 15
= –4(x2 – 6x + 9) + 36 – 15
= –4(x – 3)2 + 21
∵ Coefficient of x2 = –4 < 0
∴ The maximum value of y = –4x2 + 24x – 15 is 21 when x = 3.