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.

ƒ o G

Accepted Answer

Hello. Today we're going to be using the 2 given functions to find the following composite function and its domain. So we are told that the function a of x is equal to the absolute value of x minus 2. We are also told that the function c of x is equal to the square root of x+7. We want to use these two functions to find the composite function a of c of x. So how can we go about finding the function a of c of x? Well to make things a little bit easier, we can rewrite the notation of the composite function as a of c of x. What this means is that we are taking the function c of x and plugging it into the function a of x. What this means is that for every variable in a of x, we're going to be replacing the variable x with the function c of x. So if we plug in c into a, we can rewrite the composite function as the absolute value of the square root of x+7 minus 2. Now this function cannot be simplified any further, so this is going to be the final form of the composite function A of c of x. Now that we have the composite function, we want to find the function's domain. Now the function's domain is going to be the intersection of the function A and C. For the function A, the function is given to us in the form of a polynomial. Any polynomial does not have any restrictions on its domain, so the domain is going to be all real numbers, or in other words, the domain will be from negative infinity to positive infinity. Now let's take a look at the domain for the c of x function. Now the c of x function is in the form of a square root function. Recall that for any square root function such as the square root of x, the domain must be that x is greater than or equal to 0. What this means is that we're going to be taking the inside argument from the square root and setting it greater than or equal to 0 to find its domain. So we're going to be taking x+7, setting it greater than or equal to 0, then solving for x. In order to solve for x, we're going to subtract 7 to both sides of the inequality, and that is going to leave us with x is greater than or equal to negative 7. What this means is that we want the values of x to be negative 7 and going to positive infinity. Or in other words, the domain for c of x can be written as negative 7 including the value in its domain going to positive infinity. Now what we want to do is we want to determine the intersection between these two domains. Well, the intersection in this case is just going to be negative 7 to positive infinity because the restriction of the square root domain is that we cannot have any values that are less than negative 7. So the overall domain for the composite function will be from negative 7 to positive infinity including the value of negative 7. And with that being said, the overall answer is going to be c. 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.

ƒ o G

Key Concepts

Composite Functions

Domain of a Function

Absolute Value Function

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