In Exercises 82–84, find f + g, f - g, fg, and f/g. Determine ...

Question

In Exercises 82–84, find f + g, f - g, fg, and f/g. Determine the domain for each function. f(x) = √(x + 7), g(x) = √(x - 2)

Accepted Answer

Hello, today we are going to be solving for the function F minus G and identifying its domain. So the problem gives us two separate functions. We are given F of X which is equal to the square root of two X plus 11. And we are given G of X which is equal to the square root of X minus nine. And we want to use these two functions to find F minus G of X. So how can we start this process? Well, if we take a look at the combination function F minus G of X is telling us that we need to take the function of F of X and subtract the function G of X from it. So what we are going to do is we are going to fill in F of X minus G of X with the given functions. So F minus G of X will equal to F of X which is the square root of two X plus 11 minus G of X which is the square root of X minus nine. Now, currently, there is no further way to simplify this function. So this is going to be F minus G of X. Now, what about its domain? Well, figuring out the domain is going to be a little bit tricky because the F minus G of X function is made of the difference of two square root functions. So in order to find the domain of F minus G of X, we're going to have to first find the individual domains of F of X and G of X and the intersection of those domains will be the domain for F minus G of X. Now, if you don't remember how to find the domain of a square root function, recall that the only values we're not allowed to have within a square root are negative numbers. So the domain of a square root must always be positive or that X must be greater than or equal to zero. So what this means is that in order to find the individual domains of each square root function, we're going to take what is inside the square root function set greater than or equal to zero and solve for X. We'll go ahead and start with the domain of F of X. Now again, F of X is two X plus the square root of two X plus 11. So what we are going to do is take two X plus 11 and set it greater than or equal to zero. Now, what we are going to do is solve the inequality for X. We'll start by subtracting 11. To both sides of the inequality allowing us to rewrite the inequality as two, X is greater than equal to negative 11. Finally, we'll divide both sides of the inequality by two, leaving us with X is greater than or equal to negative 11 divided by two in interval notation. This tells us that the value of X is greater than or equal to the value of negative 11, which is going to be a bracket negative 11 divided by two and including all the values of X that come after it going to positive infinity. So this is the domain of F of X. Now, let's find the domain for G of X G of X is the square root of X minus nine. So we're going to take the inside of the square root which is X minus nine and set it greater than equal to zero and solve for X. All we'll need to do is just add nine to both sides. And that is going to give us X is greater than or equal to positive nine. So this tells us that the domain of G of X are all the values of nine going to positive infinity or in set interval notation bracket nine to positive infinity. So these are the individual domains for FFX and G of X to find the in to find the domain of F minus G of X. We'll need to find the intersection of both of these domains. What we could do is we could make a number line and plot all the values of the domains starting from the smallest value which is negative 11 divided by two going to the next biggest value which is positive nine going to all the numbers to positive infinity. Now, the common intersecting point between these two domains is going to be positive nine. And that means that the domain for F minus G of X is going to be all the values from positive nine going to positive infinity. And that will be the domain for F minus G of X. So just to summarize, we found that F minus G of X is equal to the square root of two X plus 11 minus the square root of X minus nine. And we found the individual domains of F of X and G of X. And the intersection of these two main two domains is the domain of F minus G of X. With that being said, the solution to this problem is going to be a. So I hope this video helped you understand how to approach this process. And I'll go ahead and see you all in the next video.

In Exercises 82–84, find f + g, f - g, fg, and f/g. Determine the domain for each function. f(x) = √(x + 7), g(x) = √(x - 2)

Key Concepts

Function Operations

Domain of a Function

Composite Functions

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