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

Question

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

Accepted Answer

Hello, everyone. We've been asked to determine F of G of X for the functions F of X equals the square root of two X minus six and G of X equals 18 X. Also, we need to find the domain of F of G of X. First thing I do is I rewrite it as F and then parentheses G of X. The reason I do this is because I can see it better to plug it in when I look at it and go okay. I know I need to plug G of X in for the variable in the F function. So this is like saying f of 18 x because that's what GFX was. So I know I'm going to put 18 X in for X in the F of X function. So that looks like the square root of two times 18 X minus six. And when I simplify that I have the square root of X -6 and there's nothing else I can simplify there. So that is my F of G of X. So that's part of our solution. Next, we need to find the domain to go with that, we're calling that domain is all the X values that make this a real number. We know that with a radical, we need to set what's under the radical symbol equal to zero. Because in order for this to be a real number, 36 X -6 has to be greater than or equal to zero. So instead of making it equal to zero, it's got to be greater than or equal to zero. So solving for X, I add 6 to both sides. So I have 36 x is greater than equal to six, Dividing both sides by 36, I get the X is greater than or equal to 6:36, which can be reduced to 1/6. So what that looks like an interval notation is that since it's greater than it starts at the 1/6, which is included, so it's a bracket and continues to positive infinity. So that is our domain, our domain is going to be from six, including sorry 1/6, including 1/6 through positive infinity. And that is answer choice B that's where we have our F of G of X being the square root of 36 X minus six. And the domain being from 1/6 to positive infinity. Have a nice day.

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

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