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

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

𝔂 = { -x - 2, -2 ≤ x ≤ - 1

{ x, -1 < x ≤ 1

{ -x + 2, 1 < x ≤ 2

Accepted Answer

Welcome back, everyone. In this problem, determine the domain and range of the function Y equals 2 x minus 3 when X is greater than or equal to 3 but less than or equal to 1, or minus X when X is greater than 1 but less than or equal to 2, and X2 minus 4 when X is greater than 2, but less than R equal to 4. Now notice here that this is a piecewise function, and a piecewise function is one defined by different expressions or formulas for different intervals of its domain. Each piece of the function applies to a specific range of input values, and the overall function is formed by combining these pieces based on the value of the input. So let's first try to figure out the domain based on what we're given here and then we'll try to figure out the range. Now what do we know about the domain? Let's make a note here, the domain. Well, if we look at our piecewise function, notice that we have a different domain for each piece of the function. OK? So here, using union notation, we could say that our domain is open bracket, -3, 1, closed bracket, union with open parentheses -1, 2 close bracket. Union with open bracket 24, sorry, parenthesis 2 for closed bracket, OK? So notice here that these intervals cover all X from -3 to 4. So that means then that we can write the domain as open bracket -34 close bracket. No, how about the range of the function? What do we know about the functions range? Well, for the range, we can basically think about all of the y values for each part of the domain for the piece of the function that applies, OK? So for example, starting with our first function for X, that's in the interval open bracket -3 to -1, OK. With at that point F of X equals 2 X minus 3. Since this is a linear function and it's increasing, the least value of the range will be the least value of X, where the highest value of the range will be the highest value of X. So at the least value of X-3, F of -3 would be 2 multiplied by -3 minus 3, which equals -9. On the other hand, at the highest value of, of -1 rather. Then The function is going to be 2 multiplied by -1 minus 3, which equals -5. So what does this mean? This tells us that the range, OK, for this interval is open bracket -9, -5 closed bracket. Now let's move on to our next interval. Now here, X, OK, is in the interval open bracket -12, OK. With FF X at this piece being equal to 4 minus X. What do you notice about this function? Well, no, this linear function is decreasing, OK? So that means the least value of X will be the highest value in the range and the highest value of X will be the least value in the range. So now, for our least value. Add FF -1. Oh sorry, at X equals -1, then F of -1 is 4 minus -1, which equals 5. And note that 5 is not attained since X equals -1 is is excluded, right? So let's just put that here. It's not attained just for us to remember that, OK? And no, at the highest value X equals 2. F of 2 is 4 minus 2, which equals 2, and that value is included. Thus for this one the range. It's going to be Open bracket 2. It includes 2 and 5 closed parentheses because it does not include 55 is not attained. Now let's move on to our last interval. Where X is between open parentheses 2 for closed bracket. At this point, we know that our function is equal to X2 minus 4. Now what do we know about this function? Well, this is a quadratic function and thus we know that it's increasing for X greater than 0. OK. So let's test our values at both endpoints. OK. First, at F of 2, at X equals 2, then F of 2 would be 2 quad minus 4, which equals 0, OK? And again, 0 is not attained because X equals 2 is excluded from our interval. On the other hand, for the highest value X E equals 4, F44 will be 4 square minus 4, that's 16 minus 4, which is 12. Thus, this tells us then that the range for this interval is going to be open parentheses 0, it's not attained, 12 closed bracket. Now, let's look at the ranges for all our intervals. Notice here that we have from -9 to -5, uh 2 to 55 not included, and 0 to 12, 0 not included. So if we think about it here. OK, the ranges from the 2nd uh open bracket to 5 close parenthesis is a subset of the 3rd range. OK. This is a subset of the 3rd of the 3rd range. It's inside this range. So if we think about it here, that means. The overall range. The overall range is going to be open bracket -9, up to -5, Union with parentheses 0, 12 closed bracket. This is the range of our function, and this is the domain of our function. Thanks a lot for watching everyone. I hope this video helped.

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

𝔂 = { -x - 2, -2 ≤ x ≤ - 1
{ x, -1 < x ≤ 1
{ -x + 2, 1 < x ≤ 2

Key Concepts

Domain

Range

Piecewise Function

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