Terminology

Functions in Algebra

Functions in algebra are like rules or instructions that tell us how to change a number (the input) into another number (the output). Think of it as a recipe that tells you what to do with your ingredients!

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Function Machines

Imagine you have a machine that takes a number, changes it using some steps, and then gives you a new number. This is what a function machine does!

For example, let's say you have a function machine that:

1. Divides the input by $2$,
2. Adds $7$ to the result,
3. Multiplies the final answer by $3$. If you put a number into this machine, it will follow these steps to give you the output. Function machines help us break down the steps of a function so we can easily see how the input turns into the output.

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Function Notation

Once you're comfortable with function machines, we can use a shorter way to write these rules. This is called function notation. It's like giving your function a name!

For example, if we have a function that multiplies a number by $5$ and then adds $3$, we can write it like this:

$f(x) = 5x + 3$

Here's what this means:

• $f(x)$ is the name of the function. You can think of it as the function's nickname!
• $x$ is the input (the number you start with).
• $5x + 3$ is the rule that tells you what to do with the input. So, if you put in the number $2$, the function will do this:

4. Multiply $2$ by $5$ (which gives you $10$),
5. Then add $3$ to get $13$. So, $f(2) = 13$.

Function notation is just a quicker way to write the steps, and it's very handy when you start solving problems!

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Important Points to Remember

• A function is just a rule that changes one number into another.
• A function machine helps us understand the steps of a function.
• Function notation is a quick way to write down the rule for a function.

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Terminology

Now that you understand how functions work and how to use function machines and function notation, let's move on to some important terms that are used when talking about functions: Domain, Codomain, and Range. These might sound a bit tricky at first, but with clear examples, you'll get the hang of them!

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1. Domain

The domain of a function is all the possible values that you can put into the function. In other words, it's the set of all possible inputs.

Think of the domain as the "allowed" numbers that you can use with the function. For example, if the function is $f(x) = x + 5$, you can choose any number for $x$, so the domain is all real numbers.

Example:

• If $f(x) = 2x + 3$, the domain is all the numbers you can put in for $x$ (like -$2, 0, 5$, etc.). In this case, the domain is all real numbers because you can put any number into the function and get an output.

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2. Codomain

The codomain is all the possible values that could come out of the function, even if some of them don't actually appear as outputs. It's like a big box that holds all the potential outputs.

Think of the codomain as a list of all possible outcomes, even if the function doesn't produce all of them.

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3. Range

The range is the set of all actual outputs that you get from the function after you've used all the inputs from the domain. This is sometimes called the "image" of the function.

Think of the range as the list of results you actually get when you put numbers from the domain through the function. The range is always a subset of the codomain.

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Putting It All Together:

Here's a simple way to remember the difference between domain, codomain, and range:

• The domain is what you can put into the function (the inputs).
• The codomain is the set of all possible values the function could possibly output.
• The range is what actually comes out after using the function (the outputs). Understanding these terms is important because they help you describe how functions behave and what kinds of results you can expect when using them.

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By mastering these concepts, you'll be better prepared to understand and work with functions in your Junior Cycle Maths exam. Keep practicing, and remember that these terms are just ways of organizing and thinking about the different parts of a function. You're doing great—keep it up!

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