Given functions f and g, (g∘ƒ)(x) and its domain. ƒ(x)=1/(x-2)...

Question

Given functions f and g, (g∘ƒ)(x) and its domain. ƒ(x)=1/(x-2), g(x)=1/x

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 one divided by the quantity of X minus 16 and G of X is equal to one divided by X. We also need to find the domain of G of F of X. Here we have four answer choice options, ABC and D. And from these options, we have two different expressions for G F F of X, one of which is X minus 16 and the other is X divided by the quantity of one minus 16 X. And then we have multiple different possible domains for G of F. So we want to begin by finding the domain of our first given function which is F. Well, if we recall the function F of X is equal to one divided by the quantity of X minus 16. Well, to find the domain, we just need to notice here that we have X in the denominator and we know we cannot divide by zero. So our denominator X minus 16 cannot be equal to zero, adding 16 to both sides. We see that have X be equal to 16. So with this definition here, we can write the domain of F in interval notation. So we first go from negative infinity to 16. And both of these terms are closed by parentheses. Since infinity is always excluded and 16 cannot be included because it will cause our denominator to be zero. So we go from, again, from negative infinity to 16 and then we go from to positive infinity. So we found the domain of X and we also want to identify the domain of G. Well, if we recall G of X is equal to one divided by X here, our denominator is just X and so we know X cannot be equal to zero. Now rewriting our domain in interval notation, we go from negative infinity to zero and then from zero to positive infinity. And again, all of these terms here for this interval as well are closed by parentheses. So now we want to move on to finding the expression for G of F of X. And so we can begin by rewriting this expression as G of the, the function of F of X. So G of F of X means we just need to substitute in the expression given for F of X. So we actually have G of one divided by the quantity of X minus 16. Now we just need to substitute in this expression for X from our original G of X function. So instead of G of X being one divided by X, we have G one divided by the quantity of one divided by the quantity of X minus 16. Well, one divided by a fraction. We just have to take one times the reciprocal of the fraction. So we get G of F of X or G of one divided by the quantity of X minus 16 simplifies to just X minus 16. So summarizing, we have G of F of X is equal to X -16. Now, we have found our required expression here for G of F of X. And we just need to identify the domain which is going to be the domain of the inside function, which is F of X, which earlier we identified the domain to as going from negative infinity to 16 and then from 16 to positive infinity. So with our two solutions here, we are left with answer choice C where G F F of X is X minus 16 and the domain of G F F is from negative infinity to 16 and from 16 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, (g∘ƒ)(x) and its domain. ƒ(x)=1/(x-2), g(x)=1/x

Key Concepts

Function Composition

Domain of a Function

Restrictions from Rational Functions

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