Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and ...

Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.

$ƒ (x) = \frac{4}{x}, g (x) = x + 4$

Accepted Answer

Hey, everyone here we are asked to determine G of F of X for the given functions F of X is equal to 10 divided by X and G of X is equal to X plus seven. And we are also asked to find the domain of G of F of X. So here our first step in this problem is to find the domain of our inside function which is F of X. So the domain of F can be found by looking at the expression given for F of X which is 10 divided by X. Well, here we see that X is within or an element of all real numbers are, but we see that the X is in the denominator. Well, we know we do not want to divide by a denominator of zero. So we see that X is within all real numbers except or minus the set of zero. So with this domain identified, we can go ahead and find our expression for G of F of X. Keeping in mind that the domain of G is our, so we can begin with rewriting G of F of X S G of the quantity of F of X. And so now we just need to substitute in the expression for F of X. So we have G of 10 divided by X. And now, instead of having G of X, we have G of 10 divided by X. So all we need to do from here is to substitute in 10 divided by X for X from our original expression or function of G of X. So G of 10 divided by X gives us 10 divided by X plus seven. Now, we just want to rearrange this expression here by combining both of our terms. So we see we have a whole number and a and a fraction. So we need to first find a common denominator. All we have to do is multiply our home number seven by the denominator of the fraction. So seven gets multiplied by X divided by X. Now, we just want to simplify this product here and we see our expression becomes 10 divided by X plus seven X divided by X. Now clearly we have common denominators. So we can just combine our enumerators. And we see that G of F of X or G of 10 divided by x simplifies to just 10 plus seven X all divided by X. So now we have found our expression for G of F of X, but now we need to also find the domain of this expression. Well, here we see that we cannot have a denominator of zero. So we know that X in this G of F of X cannot be equal to zero. Well, here, our inside function is F of X and so F of X needs to be in the domain of G. Well, if we recall the domain of G was our, so we have F of X is within all real numbers or are, and this is minus or excluding the set of zero. So now again, we know we cannot have X be equal to zero. And so with this expression for the domain that we have found, we can rewrite this domain in interval notation. So since we are including all real numbers, we can go from negative infinity to our excluded value here, which is zero. And we know that infinite is always excluded. So it will be closed by a parenthesis. And we know that we cannot have X be equal to zero. So zero also gets closed by a parentheses. And now we need to go in our next interval from zero to positive infinity to cover the rest of our numbers. So again, zero and infinity both get closed by parentheses. So with this, we have found our expression for G of F of X to be the quantity of 10 plus seven X divided by X. And we also have found two intervals to express our domain of G F F of X where we go from negative infinity to zero. And from zero to positive infinity. So with these two solutions, we are left here with answer choice B thanks for watching. I hope you found this video helpful and I'll see you again next time.

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = \frac{4}{x}, g (x) = x + 4$

Key Concepts

Function Composition

Domain of a Function

Rational Functions and Restrictions

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