In Exercises 19–32, find the (a) domain and (b) range.
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Question

In Exercises 19–32, find the (a) domain and (b) range.

________

𝔂 = 5 - √ x² - 2x - 3

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the domain and range for the function F of X is equal to -2 minus the square root of 9 minus X + 1 to the power of 2. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. We're asked to solve for the domain, which is our first answer, and solve for the range, which is our second answer, for this particular function of F of X. So now that we know that we're ultimately trying to solve for the domain and range of this specific function of F of X, let us first off focus our attention on solving for the domain. And then we'll solve for the range, and once again, let's note that we are solving for the domain and range of the function F of X is equal to -2 minus the square root of 9 minus X + 1 to the power of 2. Fantastic. So now solving for the domain, we need to note that for the square root to be real, as we should recall, 9 minus X + 1 to the power 2 has to be greater than or equal to 0. So that leads to -3 is less than or equal to X plus 1, which is less than or equal to 3, which will mean that -4 is. less than or equal to X and X is less than or equal to 2. So therefore, the domain has to be X is an element of the closed interval of -4.2, and a closed interval means that it's in between square brackets because a square bracket indicates a closed interval, whereas a set of parenthesis indicates an open interval. So once again, X is an element of the closed interval of -4.2. So it's gonna be square bracket 4.2 square bracket. Fantastic, and that is our answer for the domain. Hooray, we did it. Now we can solve for the range. So, as we should recall a note, we need to note that the root term, which our root term is 9 minus X + 1 to the power of 2, will range or ranges from 023. Fantastic. So from 0 to 3. Fantastic. So when the square root of 9 minus X + 12 equals 0, that will mean that at X equals -4 or at X equals 2, that will mean that F of X is going to be equal to -2 minus 0, which is going to give us an answer of -2. And once again, This is for the condition where our square root. Of 9 minus X + 1 to the 2 is going to be equal to 0. So in red, I'll make a note here that this is equal equal to 0. So let's make a note of this in parenthesis then quotation marks, so equal to 0. Fantastic. So when our next step is when our square root is going to be equal to 3, so when the square root of 9 minus X + 1 to the power 2 is equal to 3. That will mean that at X equals -1, that will mean that when X is equal to -1, that will mean that F of X is going to be equal to -2 minus 3, which is going to be equal to -5. So therefore, without a doubt, the range will be F of X is an element of the closed interval of -5.2, or square 5.2 square bracket. And that's it. We've officially solved for this problem. Hooray, we did it. We solved for the domain and range. So to quickly recap here, our final two answers without a doubt have to be the domain is a closed interval of square bracket 4.2 square bracket, and our range is a closed interval of square bracket. Minus or -5.2 square bracket. And that's it, we've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 19–32, find the (a) domain and (b) range.
________
𝔂 = 5 - √ x² - 2x - 3

Key Concepts

Domain of a Function

Range of a Function

Quadratic Expressions

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