In Exercises 31–50, find f−g and determine the domain for each fu...

Question

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

Accepted Answer

Hello everyone in this video, we're going to be looking at this practice problem where we want to find f minus three and write the domain and were given that F of X is equal to the square root of x minus seven and G of X is equal to the square root of X plus 1.5. So in this case we need to figure out what F minus G is. So F minus G represents f of x minus G of X. So seeing this, we can plug the values of F of X and G of X into our equation to get f minus G. So if we do that, f of X is equal to the square root of X - minus G of X which is equal to the square root of X plus 1.5. So now we've written of minus G. But we also need to find the domain and recall that the domain is the range of X values. That will result in this equation giving us a real number. So we want to make sure that there's no imaginary numbers in this range. And where can imaginary numbers come up? Well in this case because we have square roots recall that if we have a negative square root that can lead to the imaginary component I. So we're trying to avoid having negative numbers in the square roots which means that we want the numbers inside the square root to be greater than or equal to zero. So he can express that as inequalities where we want x minus seven to be greater than or equal to zero and we also on X plus 1.5 to be greater than or equal to zero. And now we can solve this first inequality by adding seven Leaving us with X greater than or equal to seven. And if I do the same to my second inequality, I'll subtract 1.5 from both sides. Leaving me with X has to be greater than or equal to negative 1.5. So now you can see that I have two ranges for X. So I want to make sure that I find the range that will ensure that both of my square roots will be positive. So if I draw a number line, I'll say that This is 1.50 and this is seven. So my first Range X greater than or equal to seven is saying that There's a bracket at seven because of the equal to and X has to go to the right sole draw an arrow going to the right. So it's saying that from seven upward My square root X -7 will be positive. And if I do that for my second inequality X greater than equal to negative 1.5, I'll do a bracket at the 1.5 mark. And again I need to go to the right because it's saying X greater than or equal to negative 1.5. So this one is saying that anywhere above negative 1.5 X plus five X plus 1.5 will be positive. So in this case you can see that the first range is missing from the negative 1.5 to the seven range. And what that means is that in this red section or from negative 1. to 7, The equation square root X -7 will result in a negative number, which is what we're trying to avoid. So we want to include only This range here from seven upward, which means we want The interval 2 positive infinity. And this becomes our domain because this is the range of X values that will result in real numbers. So these become our final answers where f minus G is equal to the square root of x minus seven minus the square root of X plus 1.5. And the domain is seven to positive infinity. And you can see that this aligns with answer choice C. So hopefully this video helped and see you next time.

In Exercises 31–50, find f−g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

Key Concepts

Function Operations

Domain of a Function

Combining Domains

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