Piecewise-Defined Functions

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Question

Piecewise-Defined Functions

In Exercises 35 and 36, find the (a) domain and (b) range.

𝔂 = { √ -x, -4 ≤ x ≤ 0

{ √ x, 0 < x ≤ 4

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says determine the domain and range of the function, and here we're given a piece-wise function. Y is equal to the square root of the quantity of 2 + X in quantity, when minus 2 is less than or equal to X, which is less than or equal to 0, and the square root of the quantity of 2 minus X in quantity, when 0 is less than X, which is less than or equal to 2. Now we need to determine the domain and range of this piece wise function. So we're gonna start by looking at the domain. In order to determine the domain for the entire function, we need to look at the domain for each of our pieces. So, we're gonna start off by looking at the first piece, which is the square root of the quantity of 2 + X in quantity. Now, here we have a quantity underneath a radical. And so we want to do is ensure that the quantity underneath the radical is always um greater than or equal to zero. So, we're gonna set 2 + X to be greater than or equal to 0. And we're going to solve this for X. In doing so, we'll see that X has to be greater than or equal to minus 2. Now, if we look at the interval over which this piece is defined, it's going from -2 is less than or equal to X, which is less than or equal to 0. So X is always going to be greater than or equal to -2. So that means that the domain for this piece is going to be the close interval, going from -2 to 0. Now, if we look at our second piece, which is the square root of the quantity of 2 minus X in quantity. Again, we have a radical, and we want to ensure that the quantity underneath the radical is a non-negative value. So, we're gonna set 2 minus X to be greater than or equal to 0. And again, we'll solve this for X, and then we see that X has to be less than or equal to 2. And if we look at our interval over which this function is defined, which is 0 is less than X, which is less than or equal to 2, the value for X over this interval will always less than or equal to 2. So that means that the domain for this piece is going to be the same as interval, and that's going to be the left open interval, going from 0 to 2. So what we want to do now is combine our two intervals, our two domains here, to get the total domain for our function Y. And when we combine those, we get the close interval going from -2 to 2. So now we have our domain, we can look at our range. And so here we're going to do the same sort of process. We're going to look at each piece of our piece-wise function independently. So, we'll start off with our first piece, and if we look at our function, which is the square root of the quantity of 2 + X in quantity, we see that function is always increasing over our interval. And so, That means that the minimum value is going to occur when X is equal to -2, and the maximum value is going to occur when X is equal to 0. So, looking at X equal to -2, with this first piece, we're going to have the square root. Of the quantity of 2 plus minus 2 in quantity, and this is going to be equal to 0. And if we look at X equal to 0, We're going to have the square root of the quantity of 2 + 0 in quantity, which is going to be equal to the square root of 2. So the range for the first piece is going to be the close interval, going from 0 to square root of 2. So we're gonna do something similar for the second piece of our piece-wise function. So here if we look at our second piece, it's going to be the square root of the quantity of 2 minus X in quantity. And this function is always decreasing. So that means that I'm going to have a maximum value when X is equal to 0 and a minimum value when X is equal to 2. So starting off with X equal to 0. We're going to have the square roots of the quantity of 2 minus 0. In quantity, which is going to be equal to the square root of 2. And if we look at X equal to 2, we're gonna have the square root of the quantity of 2 minus 2 in quantity, which is equal to 0. But here we have to be a little bit careful. Since the domain does not include the value X equal to 0, our function is only approaching that value. So that means that we're the range for our second piece of our piecewise function is going to be the right open interval, going from 0 to the square root of 2. So, what we want to do is choose the interval that is more inclusive, and so that is going to be the close interval. Going from 0 to the square root of 2, so that is going to be our range. So, the answer to this question is that our domain is going to be the closed interval, going from -2 to 2, and the range is going to be the closed interval, going from 0 to square root of 2. So I hope that this explanation has been useful and I'll see you in the next video.

Piecewise-Defined Functions

In Exercises 35 and 36, find the (a) domain and (b) range.

𝔂 = { √ -x, -4 ≤ x ≤ 0
{ √ x, 0 < x ≤ 4

Key Concepts

Piecewise Functions

Domain

Range

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