Question

Does the table represent a function? Explain. $$ \begin{array}{|c|c|} \hline \text { Input } & \text { Output } \\ \hline 5 & 3 \\ \hline 6 & 4 \\ \hline 7 & 5 \\ \hline 8 & 6 \\ \hline \end{array} $$

Short answer

Yes, the table does represent a function as each input has exactly one output.

Step-by-step solution

Review the definition of a function

A function is a relation where every input (also known as the 'domain') has exactly one corresponding output (also known as the 'range'). This means that if we have the same input value more than once, it should always have the same output.

Analyze the values

As we analyze the values in the table, it can be observed that every input value (5, 6, 7 and 8) corresponds to exactly one output value.

Conclude

Since the table meets the criteria of a function, where each input is associated with exactly one output, the table represents a function. No repeating input values with different outputs are found in the table.

Key concepts

Definition of a Function

Understanding what a function is can be a pivotal moment in algebra. In its most basic form, a function is a special type of relation between two sets, often referred to as the input and the output. The key property of a function is that each input is related to exactly one output. This can be compared to a vending machine: If you select a particular button (input), you expect a specific item (output) to be dispensed. In mathematical terms, if we have a set of inputs (the domain) and a set of possible outputs (the range), then a function can be seen as a rule that assigns to each input exactly one output. This is unlike a general relation, where an input might correspond to multiple outputs or none at all. This concept is foundational for understanding more complex topics in algebra and beyond.

The algebraic representation of a function usually takes the form of an equation like f(x) = y, where f represents the function, x is the input from the domain, and y is the assigned output from the range. The simplicity and order within functions make them incredibly useful for modeling all sorts of real-world phenomena.

Domain and Range

When we talk about functions, two terms frequently come up: 'domain' and 'range.' The domain of a function is the complete set of possible inputs for the function. Think of it as the 'x' values when looking at a graph or the values you'd plug into a function equation. For example, the domain of a function that assigns an age to every person would include every possible age that a person could be.

The range, on the other hand, is the set of all potential outputs or 'y' values, these are the results you get after applying the function's rule to the domain. If we continue with the previous example, the range would include every corresponding height that each age can be mapped to through the function. It's essential to identify the domain and range accurately because they define the scope and limitations of what the function can relate or describe, setting the stage for understanding more complex behaviors in graphs and equations.

Identifying Functions

One of the fundamental skills in algebra is the ability to identify whether a given relation qualifies as a function. To do this, one must look closely at the inputs and outputs presented. If you see that every input has one and only one output, congratulations—you're looking at a function! An easy way to check is to use the 'Vertical Line Test' on a graph, where if a vertical line cuts the graph at more than one point, then it's not a function. For a set of ordered pairs or a table, like the example given, make sure that no two pairs have the same first element but different second elements.

If you find a repeating input (an 'x' value) with different outputs ('y' values), then you need to disregard the notion of it being a function. However, keep in mind that having different inputs corresponding to the same output is perfectly fine in the world of functions. As long as the one-to-one correspondence rule is upheld—for each input, there is only one output—you have successfully identified a function.