How Can We Tell Functions from Non-Functions?

Knowing the difference between functions and non-functions is really important in algebra.

A function is a special kind of relationship between two groups of things. Each input from one group (called the domain) has exactly one output in another group (called the range).

Let’s break this down by looking at what a function is, how to recognize it, and how to spot non-functions.

What is a Function?

A function can be simply explained like this:

• Function Definition: A function takes an input from one set (let's call it set X) and gives an output from another set (let's call it set Y). Each input, or piece of X, goes to exactly one piece from Y. We often write this as $f(x) = y$, where $f$ is the function.

In plain terms, each input should have one and only one output.

Key Features of Functions

Here are some important features that describe functions:

1. One Output: For any input, there can only be one output. If one input leads to more than one output, then it’s not a function.

2. How They’re Written: We usually use letters like $f$, $g$, and $h$ to label functions. If $f$ is a function, we write $f(x)$ to show what the output is for the input $x$.

3. Graphs: You can also see functions in a graph. To check if something is a function, we can use the Vertical Line Test. If you draw a vertical line and it hits the graph at more than one spot, then it's not a function.

How to Spot Functions vs. Non-Functions

Here are some easy ways to tell if something is a function or not:

A. Math Check

To figure out if a relation (a way of connecting inputs and outputs) is a function:

• Mapping: Make a list of the inputs and their outputs. If any input has more than one output, then it's not a function.

For example:

• Look at this set: $R = \{(1, 2), (2, 3), (1, 4)\}$. The input $1$ gives both $2$ and $4$, so $R$ is not a function.

B. Graph Test

You can use the Vertical Line Test:

• Draw vertical lines through the graph:
  • If a vertical line crosses the graph at two or more points, it’s not a function.

For example:

• A curve like $y = x^2$ passes the test because it only hits vertical lines once. But a circle, like $x^2 + y^2 = r^2$, fails because vertical lines can touch it in two spots for some $x$ values.

C. Function Notation

Check how a function is written:

• A function should clearly show just one output for each input. If it seems confusing, you might need to look closer.

For example:

• If we write $f(x) = 3$, this is a function because it always gives the output $3$ no matter what the input is. But $y^2 = x$ for $x \geq 0$ is not a function. Here, for every positive $x$, there are two possible $y$ values (one positive and one negative).

Conclusion

In short, telling functions apart from non-functions is about understanding what a function is, looking for its unique features, and using math checks, graph tests, and proper notation.

By learning these ideas, students can easily spot functions and understand their properties. This is a crucial step for diving deeper into algebra. Understanding functions not only helps in math but also in solving problems in everyday life.