1. Domain
and
Range

2. Domain: In a set of ordered pairs, (x, y), the domain is the set of all x-coordinates (independent values).
Range: In a set of ordered pairs, (x, y), the range is the set of all y-coordinates (dependent values).

3. Closed Circles = number is included Open Circles = number not included
Interval Notation
(a,b) implies a< x < b
(a , b] implies a< x ≤ b x x a b a b
Closed Circles = number is included
Open Circles = number not included

4. that direction infinitely
Interval Notation
(a,∞) implies x > a
[a , ∞) implies x ≥ a x x a a
An arrow at the end of
the line means the
lines continues in
that direction infinitely

5. Interval Notation (-∞,b) implies x < b (- ∞ , b] implies x ≤ b b b

6. Closed Circles = number is included Open Circles = number not included
Interval Notation
[a,b) implies a ≤ x < b
[a , b] implies a ≤ x ≤ b x x a b a b
Closed Circles = number is included
Open Circles = number not included

7. Ex:{(2,3),(-1,0),(2,-5),(0,-3)} Domain: {2,-1,0} Range: {3,0,-5,-3}
The set of ordered pairs may be a limited number of points.
Given the following set of ordered pairs, find the domain and range.
Ex:{(2,3),(-1,0),(2,-5),(0,-3)}
If a number occurs more than once, you do not need to list it more than one time.
Domain:
{2,-1,0}
Range:
{3,0,-5,-3}

8. The set of ordered pairs may be an infinite number of points, described by a graph.
Given the following graph, find the domain and range.

9. Range = [0,∞)
Domain = R

10. Example: Domain: {x: x≥5} or [5,∞) Range: {y: y≥0} or [0, ∞)
The set of ordered pairs may be an infinite number of points, described by an algebraic expression.
Given the following function, find the domain and range.
Example:
Domain:
{x: x≥5} or [5,∞)
Range:
{y: y≥0} or [0, ∞)

11. Practice: Find the domain and range of the following sets of ordered pairs.
1. {(3,7),(-3,7),(7,-2),(-8,-5),(0,-1)}
Domain:{3,-3,7,-8,0}
Range:{7,-2,-5,-1}

12. 2.
Domain= [3, ∞)
Range: R

13. 3.
Domain={all reals}
Range:{y:y≥-4} 4.
Domain={x:x≠0}
Range:{y:y≠0}

14. 5. Domain={x: -2≤x≤2} Range:{y: -2≤y≤2} 6. Domain={all reals}
Note: This is NOT a Function! 5.
Domain={x: -2≤x≤2}
Range:{y: -2≤y≤2} 6.
Domain={all reals}
Range:{all reals}

15. Types Of Relations (including functions)
Chapter 4D

16. One – to – One relations
A one to one relation exists if, for any x-value, there is a corresponding y-value and vice versa.
e.g. [(1,1), (2,2), (3,3), (4,4)]

17. One – to – many relations
A one to many relation exists if, for any x-value, there is more than one y-value.
But for any y – value there is only one x – value
e.g. [(1,1), (1,2), (3,3), (3,4)]

18. Many – to – one relations
A many to one relation exists if, for any y-value, there is more than one x-value.
But for any x – value there is only one y – value
e.g. [(-1,1), (0,1), (1,2)]

19. Many – to – many relations
A many to many relation exists if, there is more than one x-value and y – value.
e.g. [(0,-1), (0,1), (1,0), (-1,0)]

20. Functions One to one or many to one are called functions
Use a vertical line test to test the theory
Vertical line cannot intersect the curve/line more than once

21. Function Notation
Chapter 4F

22. What is function notation????
Consider y = 2x
The y value is determined by the x values
Therefore as a results y is a function of x!!!!
This is where simple substitution comes into play (“,)

23. f(x) = x2 – 3 f(-2) f(2a) f(x) = x2 – 3 f(1) = (-2)2 – 3 f(1) = 4 – 3
F(2a) = (2a)2 – 3
f(1) = 22 a2 – 3
f(1) = 4a2 – 3

24. Full function Notation
A . { (x, y): y = x2 – 4}
f : X Y, f(x) = …….
f : R R, f(x) = x2 – 4
B . y = 3x – 4, -2≤ x ≤ 5
f : [-2 , 5] R, f(x) = 3x – 4
C . y =
f : R\ {0} R, f(x) =
Domain CoDomain Rule

25. Special types of Functions ( including hybrid functions)
Chapter 4G

26. Recap!! One to one or many to one are called functions
Which we already know!!!

27. When x = 0 & x = 3, y = 1 therefore not a 1 to 1 func.
Correct or Wrong
Which of the Follwing demonstrates a one to one relationship???
A. {(0, 1), (1, 2), (2, 3), (3, 1)}
B . {(2, 3), (3, 5), (4, 7)}
C . f(x) = 3x
When x = 0 & x = 3, y = 1 therefore not a 1 to 1 func.
There is only one y value for each x value therefore this is a 1 to 1 func.
From sketching this graph you can clearly see that it is a linear graph therefore this is a 1 to 1 func.

28. Restricted functions
Restricted functions can be placed on a function through it’s domain.
E.g. f(x) = (x + 1)2
We can make the func 1 to one by splitting it into two separate domains:
( -∞, -1] or [-1, ∞)

29. Hybrid Functions
If sketching by hand I would find x and y intercepts as these two equations represent linear graphs
A hybrid function has many different subsets and separate domains.
Do worked example 20 with them on the board on page 189.

30. Power Functions (hyperbola, truncus and square root functions)
Section 4E page 175

31. What we all ready know! The shape of a Quadratic and a Cubic.
What Shape are they when they are +?
What shape are they when the equation has a – in front of the x2 or x3?
All of the functions that you have leant so far are known as power functions.

33. New Power Functions!!
When n = -1, f(x) = x-1 this is known as a HYPERBOLA
When n = -2, f(x) = x-2 this is known as a TRUNCUS
When n = 1/2, f(x) = x1/2 this is known as a Square root func.
General Power Function Form:
f(x) = a(x – b)n + c

34. The Hyberbola

35. The Hyberbola
The graph shown is called a hyperbola, and is given by the equation y = 1/x
This can also be represented as the power function y = x −1.
The graph exhibits asymptotic behaviour.
That is, as x becomes very large, the graph approaches the x-axis but never touches it. As x becomes very small (approaches 0), the graph approaches the y-axis, but never touches it.
So the line x = 0 (the y-axis) is a vertical asymptote, and the line y = 0 (the x-axis) is the horizontal asymptote.
Both the domain and the range of the function are all real numbers, except 0; that is, R \ {0}.
The graph of y = x −1 can be subject to a number of transformations.
Consider y = a(x −b)−1 + c.

36. Dilation Factor y = a(x −b)−1 + c.
The value a is a dilation factor. It dilates the graph from the x-axis.
For example, the graph of y = compared to the basic graph of y = is shown in the graph on the left.

37. Reflection y = − a(x −b)−1 + c.
If a is negative, the graph of the basic hyperbola is reflected in the x-axis.
If x is replaced with −x, the graph of the basic hyperbola is reflected in the y-axis.
For example, the graphs of y = (x +1)−1 and y = − (x +1)−1 are reflections of each other across the y-axis

38. Translation Horizontal Translation
For example, the graph with equation y = (x +1)−1 is a basic hyperbola translated one unit to the left, since b = −1. This graph has a vertical asymptote of x = −1 and domain R \ {−1}, and a horizontal asymptote of y = 0.
If a basic hyperbola is translated one unit to the right, it becomes y = (x - 1)−1 , with a vertical asymptote of x = 1 and domain R \ {1}.

39. Translation Vertical Translation
The graph with equation y = (x)−1 - 1 is a basic hyperbola translated one unit down. This graph has a horizontal asymptote of y = −1, a range of R \ {−1} and a vertical asymptote of x = 0.
If a basic hyperbola is equation y = (x)−1 + 2 translated two units up, it becomes with a horizontal asymptote of y = 2 and a range of R \ {2}.

40. Translation Summary (HYPERBOLA)

43. When n = -2, f(x) = x-2 this is known as a TRUNCUS
The function is undefined for x = 0. Hence, the equation of the vertical asymptote is x = 0 and the domain of the function is R \{0}.
The graph approaches the x-axis very closely but never touches it. So y = 0 is the horizontal asymptote.
Since the whole graph of the truncus is above the x-axis, its range is R+ (that is, all positive real numbers).
When n = -2, f(x) = x-2 this is known as a TRUNCUS

44. Dilation

45. Reflection

46. Horizontal Translation
Vertical Translation

47. Translation Summary (TRUNCUS)

50. ON YOUR CAS CALC

51. The Square Root Function

52. Reflection
If a is negative, the graph of a basic square root curve is reflected in the x-axis. The range becomes (−∞, 0]. The domain is still [0, ∞).

53. Reflection

54. Horizontal Translation

55. Vertical Translation

56. Translation Summary (Square Root Function)

58. On CAS Calc