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

Question

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

$ƒ (x) = √ (x - 1), g (x) = 3 x$

Accepted Answer

Hey, everyone here we are asked to determine G F F of X for the given functions F of X is equal to the square root of the quantity of two X minus six and G of X is equal to 18 X. And we are also asked to find the domain of G of F of X. So he here we have four answer choice options. A B C and D each with a different expression for G of F of X. And they also include a different domain for G of F. So we can begin by finding our domain for our inside function which is F of X. So the domain of F, the domain of F can be found by looking at the expression given for F of X. So we see we have the square root of two X minus six. So we know that we cannot take the square root of a negative. So the expression underneath the root must be greater than, or equal to zero. So we can set this expression greater than or equal to zero. And we have again that two X minus six is greater than or equal to zero. Now just solving for X, we first add six to both sides and we have two, X is greater than or equal to six. And now solving for X, we need to get rid of the coefficient by dividing both sides by two. And we see that the domain of F is where X is greater than or equal to three. So now with this inequality identified, we see that X is within or an element of all real numbers when X is greater than or equal to three. So we also need to know that the domain of G is going to be our or all real numbers. So G is the domain of G is all real numbers. And so now we can move on to finding our expression for G of F of X. And we can first rewrite this expression as G of the quantity of F of X. And so now we just need to substitute in the given expression for F of X and we have G of the square root of two X minus six. And so now, instead of having G of X as our given function is we now have G of the square root of two X minus six. So we just need to substitute in this expression for X from our original fun of G F X. So G F X now becomes G of the square root of two X minus six, which gives us 18 times the square root of two X minus six. So here is our simplified form of G of F of X. And so now we need to find the domain of this expression here. Well, again, we have the same square root. So we know that the expression underneath the root which is two X minus six needs to be greater than or equal to zero because we cannot take the square root of a negative. So again, we know this inequality is also equivalent to X being greater than or equal to positive three. So now we also know that F of X should be in the domain of G because F of X is the inside function in this expression. So F of X is within all real numbers and we also need X to be greater than or equal to three. So with these two statements here, we can rewrite our domain of G of F as an in interval notation. So we again go from three and go to positive infinity to include the all real numbers. And now we know infinity is always excluded. So it gets closed by a parentheses. And then three, we know is included because of our non strict inequality. So three gets closed by a bracket. So with this, we have found our domain of G of F going from three included to positive infinity. And we also found our expression for G of F of X to be 18 times the square root of two X minus six. So with these two solutions, we are left here with answer choice. C Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7.
$ƒ (x) = √ (x - 1), g (x) = 3 x$

Key Concepts

Function Composition

Domain of a Function

Square Root Function Domain Restrictions

Watch next