Domain and Range

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).

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

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

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

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

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}

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.

Range = [0,∞) Domain = R

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, ∞)

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}

2. Domain= [3, ∞) Range: R

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

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}

Types Of Relations (including functions) Chapter 4D

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)]

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)]

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)]

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)]

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

Function Notation Chapter 4F

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 (“,)

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

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

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

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

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.

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, ∞)

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.

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

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.

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

The Hyberbola

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.

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.

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

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}.

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}.

Translation Summary (HYPERBOLA)

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

Dilation

Reflection

Horizontal Translation Vertical Translation

Translation Summary (TRUNCUS)

ON YOUR CAS CALC

The Square Root Function

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, ∞).

Reflection

Horizontal Translation

Vertical Translation

Translation Summary (Square Root Function)

On CAS Calc