More composite functions Let ƒ(x)...

Question

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

G o G

Accepted Answer

Hello. Today we're going to be finding the following composite function with the given function c of x. So we are told that the function c of x is equal to the square root of x+7. We want to use this function to find the composite function c of c of x. Now, we can go ahead and rewrite the composite function as c of c of x. What this means is that the function c is the input value for the function c, or in other words, we're going to be inputting this function into itself. So how do we go about writing this composite function? Well, the first thing we're going to do is we're going to write the square root of x+7. However, if we're plugging in the function c into itself, that means that the square root of x+7 is going to replace the variable x. So we can rewrite the composite function as the square root of the square root of x+7+ 7. This is going to be the value for the composite function C of c of x. Now what we need to do is we need to find the domain of this function. Now in order to find the domain of the function, we need to recall how we take the domain of a square root function. Recall that if you have the function in the form of the square root of x, the domain must be that x is greater than or equal to 0. So in the composite function, the entire function is a square root function. However, we have a square root function inside the square root function. So that means the domain is going to be the intersection between the square root of x+7 and the square root of x+7. So what we're going to do is we're going to take the square root of x+7, take the inside argument which is x+7, and set it greater than or equal to 0. Then we want to solve for x. There's only one step we need to do to solve for x and that is to subtract 7 to both sides of the inequality. That will leave us with x is greater than or equal to negative 7. This is going to be the domain for the first instance of x, the square root of x plus 7. However, if we were to take the innermost square root which is also the square root of x+7, we would end up with the same domain which is x greater than or equal to negative 7. So for the composite function, the domain is going to be the intersection of these two domains. However, since both of the domains are exactly the same that means the composite function's domain is going to be the values of x must be greater than or equal to 7 Or in set builder notation, we're going to have the domain start at negative 7 and include the value, and we want all the values greater than negative 7 that go to negative infinity. This will be the domain for the composite function. So just to recap, the composite function is the square root of the square root of x+7+7 and the domain is going to start from negative 7 including that value continuing to positive infinity. And with that being said, the answer to this problem is going to be b. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

G o G

Key Concepts

Composite Functions

Domain of a Function

Absolute Value Function

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