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

Question

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

$ƒ (x) = \sqrt (x + 2), g (x) = - (\frac{1}{x})$

Accepted Answer

Hey, everyone here we are asked to determine F of G FX for the given functions F of X is equal to the square root of the quantity of X plus 12 and G of X is equal to 14 divided by X. We are also asked to find the domain of F of G of X. Here we have four answer choice options. ABC and D each with expression for F of G FX as well as the domain of F of G. And for the expression for F of G FX, there are two different options. One is the square root of the quantity of 14 plus X all divided by X or 14 divided by the square root of the quantity of X plus 12. And then there are multiple different possibilities for the domain. So to begin this problem, we first want to find the domain of F and G individually. So starting with the domain of F again, recalling that F of X is equal to the square root of the quantity of X plus 12. We can begin finding the domain by noticing we have the expression X plus 12 underneath a radical So we know this expression must be greater than or equal to zero since we cannot take the square root of a negative. So we have X plus 12 is greater than or equal to zero. And now we can solve for X by subtracting 12 from both sides of the inequality. And we get that X must be greater than or equal to negative 12. So now moving on to G FX, we have G FX is equal to 14 divided by X. Well, we have X in the denominator and we know we cannot divide by zero. Therefore, X cannot be equal to zero. So we can write the domain as from negative infinity to zero and from zero to positive infinity and infinity is always excluded and therefore closed by parentheses. And here we, we cannot have X B equal to zero. So zero is also excluded and closed by parentheses. So now that we have individually determined the domains of F and G, we can move on to assessing F of G of X and we first need to find the expression for F of G FX. And so we can rewrite this expression as F of the quantity of G FX. And now we can just substitute in G, the expression for G of X. So we actually have F of 14 divided by X. So now we can clearly see we just need to substitute in 14 divided by X four X from our original F of X function. And so we get the new expression where we have the square root of 14 divided by X plus 12. And so now combining the terms that are underneath the radical, we find that our expression for F of G of X simplifies to the square root of the quantity of 14 plus 12 X all divided by X. So now we have found the expression for F of G FX. So the next thing we need to do is find the domain of this expression here. So the first thing we can notice is we have an X in the denominator and we know that then X cannot be equal to zero because we cannot divide by zero. So we have to include that in our domain. And we also need to notice that the inside function is G FX. Therefore, G FX should be in the domain of F. So we also have to keep in mind the domain of G. So now the last thing we need to notice with our expression is that we have an expression underneath the radical, which means we take this expression which is 14 plus 12 X all divided by X and set it to be greater than or equal to zero. Since again, we cannot take the square root of a negative. So now we need to just solve for X by first multiple, both sides by X. And so we get 14 plus 12 X is greater than or equal to zero. And now continuing to solve for X, we just subtract 14 from both sides and we get 12, X is greater than or equal to negative 14. Now getting X completely alone, we divide both sides by 12 and we get that X is greater than or equal to negative seven divided by six. So now knowing that the domain of G must be within the domain of F as well as that X cannot be equal to zero and that it must be at least greater than or equal to negative 76. We can now write our domain of F of G. So our domain of F of G becomes negative 76 to 0 and from zero to positive infinity, now infinity and zero are excluded and therefore closed by parentheses. But negative 76 is included due to our inequality that we just simplified. So negative 76 gets closed by a bracket. So now with our expression for F of G FX, as well as the domain identified, we are left with answer choice. A where again, the expression for F of G of X is the square root of the quantity of 14 plus 12 X all divided by X. And the domain of F O G is from negative included to zero and from zero to positive infinity. 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. See Examples 6 and 7.
$ƒ (x) = \sqrt (x + 2), g (x) = - (\frac{1}{x})$

Key Concepts

Function Composition

Domain of a Function

Square Root and Rational Function Restrictions

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