Given the functions f(x)=x2f(x) = x^2f(x)=x2 and g(x)=x−8g(x)=\sqrt{x-8}g(x)=x−8 find (f∘g)(x)(f∘g)(x)(f∘g)(x) and determine its domain.

( f ∘ g ) ( x ) = x − 8 (f∘g)(x)=x-8 ( f ∘ g ) ( x ) = x − 8 ; D o m : [ 8 , ∞ ) Dom:[8,∞) Do m : [ 8 , ∞ )

Given the functions f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 and g ( x ) = x − 8 g(x)=\sqrt{x-8} g ( x ) = x − 8 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> find ( f ∘ g ) ( x ) (f∘g)(x) ( f ∘ g ) ( x ) and determine its domain.

give this problem a try. So in this problem, we have 2 functions. We have f of x is equal to x squared, and g of x is equal to the square root of x minus 8. We're asked to find f of g of x and determine its domain. So let's see how we can go about solving this. Well, first off, what we're asked to do is find f of g of x, and in this case, g of x is going to be the inside function. Now to do this, we can take our function for g of x and we can replace the x in our function f of x with that function. So So what this means is we're going to take this function here the square root of x minus 8, and we'll replace it with the x that we have here, so we'll have square root of x minus 8 squared as f of g of x. Now notice how I can cancel the square with the square root, giving us the function x minus 8, and this right here is going to be our function for f of g of x. So the solution for the composed function is this. But we're also asked to find the domain of this function, and when finding the domain, recall that what you first have to do is find the domain of the inside function g of x. Once you've found the domain there, you're going to find the total domain for f of g of x and then the combination of those domains is going to be our total domain. So let's start with finding the restrictions for g of x. For g of x, our function is the square root of x minus 8, and since we have a square root, nothing underneath the square root can be negative, so that means this has to be a positive number. So we can say that x minus 8 has to be greater than or equal to 0. Now what I can do from here is solve for x by adding 8 to both sides of this equation, this will get the eights to cancel on the left, giving us that x has to be greater than or equal to positive 8. So this right here is our restriction for x when it comes to our function g. Now if I go ahead and look at our function f of g of x, there are really no restrictions to this because we don't have a square root and we don't have a rational situation where this is in the denominator of a fraction, we just kind of have have x by itself minus 8. So there are no restrictions for this function, so what that means is that the total domain is going to be all real numbers from 8 to positive infinity. Basically, we can have any real number as long as it is above 8. Now you might have looked at this and thought that there were no restrictions for f of g of x, but keep in mind, you always have to pay attention to whatever the restrictions are on the inside function. So that is how you can solve these types of problems. Thanks for watching.

Given the functions f ( x ) = x 2 f(x) = x^2 f ( x ) = x 2 and g ( x ) = x − 8 g(x)=\sqrt{x-8} g ( x ) = x − 8 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> find ( f ∘ g ) ( x ) (f∘g)(x) ( f ∘ g ) ( x ) and determine its domain.

( f ∘ g ) ( x ) = x − 8 (f∘g)(x)=x-8 ( f ∘ g ) ( x ) = x − 8 ; D o m : [ 8 , ∞ ) Dom:[8,∞) Do m : [ 8 , ∞ )

( f ∘ g ) ( x ) = x − 8 (f∘g)(x)=x-8 ( f ∘ g ) ( x ) = x − 8 ; D o m : ( − ∞ , ∞ ) Dom:(-\infty,\infty) Do m : ( − ∞ , ∞ )

( f ∘ g ) ( x ) = x 2 − 8 (f\circ g)(x)=\sqrt{x^2-8} ( f ∘ g ) ( x ) = x 2 − 8 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> ; D o m : ( − ∞ , ∞ ) Dom:(-∞,∞) Do m : ( − ∞ , ∞ )

( f ∘ g ) ( x ) = x 2 − 8 (f\circ g)(x)=\sqrt{x^2-8} ( f ∘ g ) ( x ) = x 2 − 8 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> ; D o m : [ 8 , ∞ ) Dom:[8,∞) Do m : [ 8 , ∞ )

3. Functions & Graphs

Function Composition

Precalculus

3. Functions & Graphs

Function Composition

Struggling with Precalculus?

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Multiple Choice

Given the functions $f(x) = x^2$ and $g(x)=\sqrt{x-8}$ find $(f∘g)(x)$ and determine its domain.

$(f∘g)(x)=x-8$ ; $Dom:(-\infty,\infty)$

$(f\circ g)(x)=\sqrt{x^2-8}$ ; $Dom:(-∞,∞)$

$(f∘g)(x)=x-8$ ; $Dom:[8,∞)$

$(f\circ g)(x)=\sqrt{x^2-8}$ ; $Dom:[8,∞)$

Verified step by step guidance

Understand the composition of functions: (f∘g)(x) means f(g(x)). This means you first apply g(x) and then apply f to the result of g(x).

Given f(x) = x^2 and g(x) = \sqrt{x-8}, substitute g(x) into f(x) to find (f∘g)(x). This gives us f(g(x)) = f(\sqrt{x-8}).

Substitute \sqrt{x-8} into f(x) = x^2, resulting in (\sqrt{x-8})^2. Simplify this expression to get x - 8.

Determine the domain of (f∘g)(x). Since g(x) = \sqrt{x-8}, the expression under the square root, x-8, must be greater than or equal to 0. Therefore, x ≥ 8.

The domain of (f∘g)(x) is [8, ∞) because x must be at least 8 for the square root to be defined, and there are no further restrictions from f(x) = x^2.

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