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. Today we are asked to find the function F over G and write its domain. So the given F of X function is the square root of X plus seven. And the given G of X function is the square root of x minus 13. We need to go ahead and find F over G. And what that means is we're going to take our given F of X function and divided by a given G of X function. So that's going to be the square root of X plus seven over the square root of X minus 13. And that's going to be the function F over G. Now we need to go ahead and find its domain with this given function, there are two possible parts to the domain for the numerator. We have a square root function and the square root function states that the domain has to be greater than or equal to zero. So we're going to take X plus seven and set a greater than or equal to zero. And for the denominator the same rule follows. However, we cannot have the denominator as zero because zero in the denominator produces an undefined value. So we need to have X greater than zero, which means that x minus 13 needs to be greater than zero. So now we need to go ahead and solve for both parts for X plus seven greater than or equal to zero. We're going to subtract seven to both sides of the inequality and that gives us X is greater than equal to negative seven. Now, what this means is we want all the values including negative seven and anything that is bigger than negative seven. So that's going to be positive infinity And for X -13, greater than zero, we're gonna go ahead and add 13 to both sides of the inequality that's going to give us X is greater than 13. Now, what this means is we want all the values not including 13 and anything that's bigger than 13. So that's going to be infinity. Now, Both of these domains contradict each other because 13 to infinity is saying we want all the values of 13, not including 13 to infinity, but we can't have anything smaller than 13, negative seven to infinity is a smaller domain or has smaller values than 13. So we can't use this part of the domain which means that the domain to this problem is going to be 13 to infinity. So our function F over G is going to be the square root of X plus seven over the square root of x minus 13. And the domain is going to be 13 to infinity. With that being said, the answer to this problem is going to be seen. So I hope this video helps you in understanding how to solve this type of problem. And I'll go ahead and see you all in the next video

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 Division

Domain of a Function

Restrictions from Denominators

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