Give examples of three functions and three relations, using at least three different

representations (e.g., symbolic representations, graphical representations, diagrams,
tables of values, lists of ordered pairs). Use the definitions for function and relations to
justify why each of your example is a function or relation.

y = 3 x

y = x^2
y = sin x

then relations that are not functions

y^2 = x or y = x^(1/2)
two values of y for every x

y = sin^-1 (x)
for every x there is an infinite number of y values
like if x = 1/2
y = 30 deg, y = 150 degrees etc

x y
1 -1
0 0
1 1

Sure! I can provide examples of three functions and three relations using different representations:

1. Example of a function using symbolic representation:
- Function: f(x) = 2x + 3

In this case, the function is explicitly defined using symbols and an equation. Every input (x) has exactly one corresponding output (2x + 3), which satisfies the definition of a function.

2. Example of a function using a graphical representation:
- Function: y = x^2

The graph of this function is a parabola that opens upwards. For every x-coordinate, there is a unique y-coordinate, and thus, it is a function.

3. Example of a function using a table of values:
- Function: f(x) = 3x - 2

Consider the table:
| x | f(x) |
|----|------|
| -2 | -8 |
| 0 | -2 |
| 2 | 4 |

Each input value in the table has exactly one corresponding output value, satisfying the definition of a function.

Now, let's move on to three different examples of relations:

1. Example of a relation using symbolic representation:
- Relation: y = x^3

This relation is not a function because for certain values of x, there can be multiple corresponding values of y. For example, if x = 2, y can be both 8 and -8.

2. Example of a relation using a graphical representation:
- Relation: y = sin(x)

The graph of this relation is a sinusoidal wave. Here, each input value (x) is associated with a unique output value (y), so it is a function.

3. Example of a relation using a list of ordered pairs:
- Relation: R = {(1, 2), (2, 4), (3, 6), (1, -2)}

In this case, the relation is not a function because the input value 1 is associated with both the output values 2 and -2. A function should have only one output per input.

In each example, the definition of a function states that every input should have a unique output, while a relation can have multiple outputs for a given input. The representations used help demonstrate these characteristics of functions and relations.