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 o ƒ

Accepted Answer

Hello. Today we're going to be using the 3 functions to find the following composite function, then we want to find the domain of the composite function. 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 b of x is equal to 2x squared plus 7. And we are told that the function c of x is equal to the square root of x+7. We want to use these three functions to find the composite function c of b of a. So how can we use the compo the functions to find the composite function? Well, we're going to need to find the innermost composite function first. The innermost composite function is going to be b of a, and the composite function b of a can be written as b of a of x. So what this means is that we're going to be taking the function a of x and plugging it into the function b of x. When we do this, we're going to replace every variable x in the b of x function with the new function a of x. So if we do this, the function b of a of x can be written as 2 multiplied by the quantity the absolute value of x minus 2 squared plus 7. Now the absolute value of x minus 2 can just be rewritten as the quantity x minus 2. So b of a of x can be written as 2 multiplied by the quantity x minus 2 squared plus 7. Now in order to simplify this function, we're going to need to expand squared. Squared is the same thing as multiplied by x 2. And when we foil these two quantities together, the quantity that we are left with is x squared minus 4x plus 4. So we can rewrite the function as 2 multiplied by the quantity x squared minus 4x++7. The next thing we want to do is we want to distribute the value of 2 into the new trinomial. Doing so allows us to rewrite the function as 2x2-8x plus 8+7. Finally, we're going to add the constants together, which is going to be 87. 8+7 will give us the value of 15 and we are left with 2x2 8x+15. This is going to be the value for the composite function b of a of x. So now that we have this value, we can use this value to solve for the remaining portion of the composite function. Now, the remaining portion of the composite function can be written as c of b of a of x. What this means is that we're going to be taking the value of the composite function, b of a of x, and plugging it in to the function c. Now the function c is given to us as the square root of x+7. And when we plug in the composite function b of a of x into c of x, we're going to be replacing every variable x with the composite function. So c of b of a of x can be written as the square root of 2 x squared minus 8 x plus 15 and then plus the constant in the c of x function, which is going to be 7. We only have one step in simplifying this function, and that's to add the constants 15 and 7 together. 15+7 gives us the value of 22, so the composite function c of b of a of x is equal to the square root of 2 x squared minus 8 x plus 22. This is going to be the value of the composite function. So now that we have the composite function, we're going to be moving on to the second part of the problem and that is to find the domain of the function. Now the composite function is in the form of a square root function. Recall that whenever you have a function such as f of x is equal to the square root of x, the domain is going to be the value of x must be greater than or equal to 0. Or in other words, whatever is inside the square root must be set greater than or equal to 0 in order to find the restrictions of x. So for our composite function, we're going to be taking the value 2x2 8x+22 and setting it greater than or equal to 0. Then what we want to do is we want to solve for x. The first thing we can do is we can divide the entire inequality by 2. This will allow us to simplify the inequality to x squared minus 4x plus 11 greater than or equal to 0. The next thing we're going to do is we're going to subtract the constant 11 to both sides of the inequality. That will leave us with x squared minus 4x is greater than or equal to negative 11. Now, we're going to complete the square on both sides of the function. In order to complete the square on the left hand side, we're going to set a variable c equal to the middle coefficient which is negative 4 divided by 2 and square that quantity, and negative 4 divided by 2 squared will give us the value of 16. Oh, I'm sorry, negative 4 divided by 2 gives us the value of negative 2. Square that gives us the value of positive 4. So we have x squared minus 4x plus 4 greater than or equal to negative 11. And since we completed the square, we're also going to have to add the value of 4 to the right hand side of the inequality. Now x squared minus 4x+4 can be simplified to x minus 2 quantity squared, and this will be greater than or equal to negative 11+4, which will give us the value of negative 7. Now at this step, we're going to need to take the square root of both sides of the inequality. But the issue here is that we are taking the square root of a negative number on the right hand side of the inequality. This will result in an imaginary value. And since we end up with an imaginary value, that means the restriction of x is not valid. Since the restriction of x is not valid, that means there is no restriction on the current composite function. And that means the domain is going to include all values of x or in other words, the domain is going to be from negative infinity to positive infinity. This will be the domain of the composite function c of b of a of x. So just to recap, the composite function that we are looking for is the square root of 2 x squared minus 8 x plus 22, and its domain is all real numbers from negative infinity to positive infinity. And with that being said, the answer to this problem is going to be a. 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 o ƒ

Key Concepts

Composite Functions

Domain of a Function

Absolute Value Function

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