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

Question

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=x+2, g(x)=x^4+x^2-4

Accepted Answer

Hello, everyone. We've been asked to determine what G of F of X is for the functions F of X equals X plus three and G of X equals five, X to the fourth plus X minus 33. Also, we need to find the domain of G of F of X. So I like to rewrite this with parentheses, so I can rewrite it as G open parentheses. And inside of putting my FFX, I do this because I need to be able to see that I'm putting the F of X function into G. So that's like saying G of X plus three. So now I can see a little clearer that I'm going to put X plus three in for X in the G function. So what that will look like is five times X plus three to the fourth power Plus X-plus three minus 33. So I'm gonna leave the fourth power on those parentheses for now. So I'll have five times in parentheses, X plus three to the fourth power. I'm gonna drop the parentheses from X plus three in the second term. So I can combine it with the minus 33 that gives me a positive X and three plus negative 33 is negative 30. Now, I would go through and multiply X plus three to the fourth power and multiply times five. But looking quickly at the answers that were given, I'm gonna say our answer choices don't want us to do all that work. So this would be the function G of F of X given those functions. So this is G of F of X and it will equal five times X plus three to the fourth power plus X minus 30. So this is part of our solution. We also need to still find the domain. Recalling that the domain is all the X values that make this function come out to a real number. I want to think to myself, is there any X value I can put in here that will not result in a real number. So if I put a negative number in here, will this still come out to a real number? Yes, I put a positive number in here. Yes. So there are no real numbers I can plug in that don't work, which means our domain is all real numbers. So from negative infinity to positive infinity and looking at my answer choices that is answer choice. D have a nice day.

Given functions f and g, find (b)(g∘ƒ)(x) and its domain. See Examples 6 and 7. ƒ(x)=x+2, g(x)=x^4+x^2-4

Key Concepts

Function Composition

Domain of a Function

Polynomial Functions

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