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After a recap of the terms domain, range, and function, we consider functions that are defined in a piecewise fashion, with a collection of nonoverlapping domains and corresponding equations mapping the inputs to the range.
But first, we need to remember that the domain, to make in, are always the 𝑥-axis and the input values, and the range is always the 𝑦-axis and the output values. We also have to remember that a function is where every element in the domain is mapped to exactly one element in the range. So this is what a piecewise function actually looks like. And a piecewise function can be defined as a function which more than one formula is used to define the range over different pieces of the domain. So what that means is, this formula one is only part of the range. It’s only part of the function if 𝑥 is in that part of the domain, and same with formula two and formula three. Now there can be as many formulas within a piecewise function as we need to be part of the function. There’s obviously no maximum, but the minimum is of course two. Because otherwise, it’ll just be a function. Now another important thing to note, is that only one formula can be applied to one domain. And that’ll make more sense as we have a look at this next example.
The function 𝑓 of 𝑥 is defined by 𝑓 of 𝑥 is equal to five minus two 𝑥, if 𝑥 is less than one, and 𝑥 squared plus three, if 𝑥 is greater than or equal to one. Find the domain and range. So looking at these inequalities, we can see what I said just before, that only one function can be applied to one part of the domain. So we can see for the first function it’s everything less than one, and then for the second function it’s everything greater than or equal to one. They can’t both be also equal. Okay. So let’s start finding the domain and range. First thing we should do, is sketch both of the graphs individually and then combine them to one axis. I’m gonna go ahead and just combine them to one axis straightaway.
So this is the graph 𝑓 of 𝑥 equals five minus two 𝑥. And this is the graph 𝑓 of 𝑥 equals 𝑥 squared plus three. Now you’ll notice these circles. We’ve done this because we’ve said in this circle that’s coloured in, that means that function is also included in this coordinate, whereas this circle is not coloured in so therefore it’s not also included. And that’s going back to how we act on number lines.
Okay. So let’s find the domain and range. Well we know that the domain is every possible input value in the 𝑥-axis. So whatever 𝑥 could be, well we know that 𝑥 could be any real number, as 𝑥 can be any negative number from the five minus two 𝑥 function, and the 𝑥 can be any positive number from 𝑥 squared plus three. We’re also given this in the question, because we’re told that 𝑥 is going to be everything less than one and also everything greater than one. So therefore, the domain is just 𝑥 in the real set of numbers. Or we could say, every 𝑥 value from negative infinity to positive infinity, if you prefer set notation.
Now looking at the range, we know that the range are all of our output values or the 𝑦-axis. And we can see that we’re going to receive every 𝑦 value greater than three. Now that’s important to note. It’s greater than three because this coordinate here is not actually included within the function. So 𝑓 of 𝑥 is greater than three.
Okay. So just to recap, a piecewise function, after we have here, is a function which has multiple formulas to define the range over different pieces of the given domain. And we know that the domain is the 𝑥-axis, all of our input values. And the range is the 𝑦-axis, all of our output values. And what we should do when we’re given a piecewise function, is sketch it and find them.
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