Functions and Graphs

Find th...

Question

Functions and Graphs

Find the domain of y = (x + 3) / (4 − √(x² − 9)).

Accepted Answer

Welcome back, everyone. In this problem, we want to identify the domain of the function G of X equals X + 6 divided by 8 minus the square root of X2 minus 36. No. If we're going to figure out what the domain of our function is, OK, then we need to find out what would make our function undefined. In other words, what input value would make GeoIX undefined? Now in this case, as I cautioned. If we look at our denominator for this function, we can tell that there are two things that would make our function undefined. First, our function is going to be undefined if the denominator of the function. Equals 0, OK. So that's one reason that one thing that would make it undefined, one condition. The second condition, and let me put that on the right, is if the expression. The expression inside the square root. Is negative. Because if it's negative, then that is gonna give us a complex number or possibly give us a complex number that would make our function undefined. So we don't want these two conditions. So let's figure out what values would make these conditions true and then use that to help us figure out the domain of GFX. Let's start with the first condition. If the denominator is 0. If the denominator is 0, that means a minus the square root of X2 minus 36 equals 0. Then that means X2 minus 36 would be equal to 8, and if we square both sides, then X2 minus 36 equals 64. And if we add 36 to both sides, then X squared would be equal to 100. And then by squaring both sides, then X would be equal to plus r minus 10. In other words, when X equals positive or negative 10, our function would be undefined. Let's move to our second condition. The expression inside the square root is negative. Well, if we don't want that to happen. Then we would want the condition, we would want the expression inside the square root to be nonnegative, and in that case, that means we'd want X2 minus 36 to be greater than or equal to 0. If it's greater than or equal to 0, that means X2d must be greater than or equal to 36. And if that's the case, square rooting both sides, that means the absolute value of X must be greater than or equal to 6. Essentially what we're seeing here, OK, is that if we square the X value, we need to, it, we need to make sure that it gives us a result that's greater than or equal to 36, OK? In other words, Let me put 1 here. In other words, that means X must be less than or equal to -6 or greater than or equal to 6 forge ofX to be defined. So now that we know both conditions for a function to be undefined, how can we write the domain to reflect these conditions? Well, let's try to just draw a big number line here to help us make sure we understand what's going on, OK? Let's try to visualize what that might look like. So we have all our real numbers from negative infinity to positive infinity, right? And from our first condition, we know that X X cannot be equal to positive or negative 10, OK? If it's equal to either of these, that means it's going to be undefined. So on our number line here. If we make some space for positive and negative 10, OK, then we know that it can't be either of these values, so negative 10. Positive and we are open circle shows that it doesn't include any of those values. Next, we know that it has to be less than or equal to -6 or greater than or equal to 6 for it to be defined. So, again, if we put that on our number line here, OK. We just try to put it somewhere that it might work. So we know it includes these values because if it equals positive or negative 6, then the expression under the term under the square root will be 0 and we'll be able to calculate it, OK? But we know that it cannot equal anything less than that absolute value, so it cannot be any of the values in between 6 to 6. But like we said, we know that uh it includes -6 all the way up to -10 and then up to negative infinity and then it includes 6 all the way up to 10 and then to infinity. So now if we were to write this, OK, as a domain, then. That means the domain of G of X is going to be open parenthesis negative infinity to -10 closed parentheses union with parentheses 10 to -6 close bracket union with open bracket 6 to 10 closed parentheses union open parentheses 10 to infinity closed parentheses. This is going to be the domain. Of G of X. Thanks for watching everyone. I hope this video helped.

Functions and Graphs

Find the domain of y = (x + 3) / (4 − √(x² − 9)).

Key Concepts

Domain of a Function

Square Root Function

Rational Functions

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