Find (fog) if f(x)=x^3+1 and g(x)=x+1?
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To determine the result of compositions of the given function, all we need to do is to plug in the second function listed g(x) into the first function f(x).
Conclusion: We'll replace x by g(x) in the expression of f(x).
(fog)(x) = [f(g(x))] = [g(x)]^3 + 1
(fog)(x) = (x+1)^3 + 1
This sum of cubes returns the product:
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
(x+1)^3 + 1 = (x + 1 + 1)[(x+1)^2 - x - 1 + 1]
(x+1)^3 + 1 = (x+2)(x^2 + 2x + 1 - x - 1 + 1)
We'll eliminate like terms within the second pair of brackets:
(x+1)^3 + 1 = (x+2)(x^2 + x + 1)
(x+1)^3 + 1 = x^3 + x^2 + x + 2x^2 + 2x + 2
(x+1)^3 + 1 = x^3 + 3x^2 + 3x + 2
The result of combining the given functions f and g is: (fog)(x) = x^3 + 3x^2 + 3x + 2.
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