A function is a special relationship between two groups of data. In this relationship, each input connects to exactly one output. This idea is important in algebra and helps us understand harder math topics later on. To tell the difference between functions and things that aren’t functions, we need to understand some key ideas about them, especially the domain and range.

To define a function, we look at ordered pairs. These pairs have an input and its matching output. For something to be a function, it needs to follow this rule: each input can only connect to one output. If we have a group of ordered pairs like ((x_1, y_1), (x_2, y_2), (x_3, y_3), \ldots), it is a function if:

1. Unique Output: For every unique input (x_i), there is one unique output (y_i), meaning (f(x_i) = y_i).
2. No Duplicate Inputs: No two pairs can have the same input but different outputs, like ((x_i, y_i)) and ((x_j, y_j)) where (x_i = x_j) but (y_i \neq y_j).

We can use tools like mapping diagrams, graphs, and functional notation to understand functions better. If we graph points on a coordinate plane, we can check if it's a function by using the vertical line test. If a vertical line hits the graph at more than one point, the relationship is not a function. This is a simple way to check if something is a function.

Now, let's talk about domain and range. The domain is all the possible inputs, while the range includes all the possible outputs. For example, for the function from the equation (y = x^2):

• The domain is all real numbers, because you can square any real number.
• The range is only non-negative real numbers (zero or positive values) because squaring a real number can’t give you a negative.

To distinguish functions from non-functions, we can use a few methods:

• Listing Ordered Pairs: Look at a list of pairs to see if any input repeats with a different output.
• Graphing: The vertical line test helps check if a graph shows a function.
• Mapping: In a mapping diagram, every input should connect to exactly one output.

Let’s explore some examples to make these ideas clearer.

Example 1: A set of points: ({(1, 3), (2, 4), (1, 5)}).

• Here, the input (1) connects to both outputs (3) and (5).
• So, this isn’t a function.

Example 2: For the relation ({(3, 7), (4, 8), (5, 9)}), every input has one output.

• Therefore, this is a valid function.

Equations can also show functions or non-functions:

• Function: The equation (y = 2x + 3) is a function because each (x) gives one unique (y).
• Non-Function: The equation (y^2 = x) is not a function because it can produce two different (y) values (positive and negative) for one (x).

Another important thing to consider is how functions behave. Sometimes, we might think a relationship looks like a function, but it isn't. For example, with (y = \sqrt{x}), people might think it gives a negative output for some (x), but we must keep our domain in check to define it correctly.

Understanding the domain and range helps us better understand functions. For the domain, we should:

1. Identify values of (x) that make the function undefined.
2. Keep in mind that some functions can’t produce certain results, like division by zero.

For the function (f(x) = \frac{1}{x-5}), it isn't defined when (x=5). Therefore, the domain is all real numbers except (5):

$$\text{Domain: } (-\infty, 5) \cup (5, \infty)$$

The range is about what output values (y) can be based on the domain. For the previous function:

• (y) can be any value except zero, because there’s no input that will make (y) equal to zero. So, we write:

$$\text{Range: } (-\infty, 0) \cup (0, \infty)$$

In summary, knowing how to tell apart functions and non-functions is all about understanding their special traits and how inputs connect to outputs. We’ve discussed definitions, the importance of domain and range, and different ways to check if something is a function. By practicing these skills, students can build a strong foundation in recognizing and working with functions in their algebra classes.

In conclusion, figuring out whether something is a function involves using visuals like the vertical line test, carefully analyzing ordered pairs, and understanding the right domain and range. Mastering these ideas is very important for growing in algebra and helps prepare you for more advanced math topics.