What is the solution of fog(x) = 0 if f(x) = x^2 + x + 1 and g(x) = x - 1?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function f(x) = x^2 + x + 1 and g(x) = x - 1.

fog(x) = f(g(x))

= f(x - 1)

= (x - 1)^2 + (x - 1) + 1

fog(x) = 0

=> (x - 1)^2 + (x - 1) + 1 = 0

=> x^2 - 2x + 1 + x - 1 + 1 = 0

=> x^2 - x + 1 = 0

x1 = (1 + sqrt(1 - 4))/2 = 1/2 + (i*sqrt 3)/2

x2 = 1/2 - (i*sqrt 3)/2

The solution of fog(x) = 0 is x = 1/2 + (i*sqrt 3)/2 and x = 1/2 - (i*sqrt 3)/2

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