Given functions f and g, find (a)

Question

(ƒ \circ g) (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 F of G of X 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 F of G of X. I like to rewrite this with parentheses. So I like to say it's F of X. So I'm opening my parentheses and then inside the parentheses, I'm putting G of X. This helps me visualize where I'm plugging things in because now I know that the value of G of X, which is the five X to the fourth plus X minus 33 is what I'm putting in for X in the F function. So I'm going to do that and that will give me I'm gonna keep it in parentheses for now, five X to the fourth plus X minus 33 then plus three. Now looking at it quickly, do I have any reason to keep it in the parentheses? I do not. So I'm gonna drop my parentheses and then look for like terms. And when I do that, I end up with five X to the fourth plus X. And my only like terms are negative 33 3 will give me a negative 30. There are no other like terms, nothing else I can combine. So F of G of X equals five X to the fourth plus X minus 30. So this is part of our solution. We're not done quite yet as we need to find the domain. We're calling it that the domain is the X values that make the function come out to a real number. I'm going to quickly think through this saying, okay, if I put a negative number in for X, will I get a real number answer? I will. If I put a positive number in, will I get an answer of real number answer? I will? Since there is no fraction and there's no radical. Our domain here works out to be all real numbers. So it goes from negative infinity to positive infinity. So when we look for the function of F of G of X to be five X to the fourth plus X minus 30 with a domain of negative infinity to positive infinity, that is answer choice B have a nice day.

Given functions f and g, find (a) $(ƒ \circ g) (x)$ and its domain. See Examples 6 and 7.
$ƒ (x) = x + 2, g (x) = x^{4} + x^{2} - 4$

Key Concepts

Function Composition

Polynomial Functions

Domain of Composite Functions

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