In Exercises 31–50, find fg and determine the domain for each fun...

Question

In Exercises 31–50, find fg 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 times G and write the domain of F times G. And we are given that F of X is equal to the square root of X plus seven and G f x is equal to the square root of x minus 13. So in this case our first step is going to find what G F times G is. And in this case F times G is just F of X times G of X. So if we rewrite that with the functions that were given, we know that F of X is equal to the square root of X plus seven And GFX is equal to the square root of X -13. So now you can see that F times G is equal to the square root of X plus seven times the square root of x minus 13. So because we can't simplify this expression or this equation any longer, we're going to figure out the domain and recall that the domain is the range of X values. That will ensure that this equation gives us a real number. So what that means is that we don't want any imaginary components. And because we're working with square roots, recall that the square root of a negative number will result in an imaginary component. So that's what we're trying to avoid. So we want to make sure that the number insider square root remains above or equal to zero. So we can express that by doing an inequality where we have X plus seven needs to be greater than or equal to zero And X -13 also needs to be greater than or equal to zero. So now solving my first inequality, I can subtract seven from both sides, leaving me with X greater than or equal to negative seven. And that simplifies that inequality. So I can go on to my second inequality And I'll add 13 to both sides, leaving me with X greater than or equal to 13. And here you can see that I have two ranges for the value of X. So I want to find the range that encompasses both of them. So I'm gonna draw a number line and I'll say that this point over here is negative seven, this point is zero and this point is 13. So if I look at my first inequality, this is saying that X has to be greater than or equal to negative seven. So I can draw a bracket and an arrow going to the right because X needs to be Greater than -7. And if I look at my second inequality, X greater than or equal to 13, I can draw a brocket at 13 and X again needs to be greater than 13. So we'll draw another arrow going to the right. So what these ranges are telling me is that From -7 to infinity? The inequality or the square root with X plus seven is going to be positive, however, from 13 to infinity, the square root x minus 13 is going to be positive. So you can see that this range here doesn't allow The Square Root X -13 to be positive. So we want to include only the second inequalities range In order to ensure that our 2nd square root also remains positive. So if we write that range, It would be 13 to positive infinity. So this becomes our domain. So these are our final answers where F of G is equal to the square root of X plus seven times the square root of x minus 13 and the domain is 13 to positive infinity. And you can see that that aligns with answer choice B. So hopefully this video helped and see you next time.

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

Key Concepts

Function Composition

Domain of a Function

Square Root Functions

Watch next