f(x) = 2x + a, g(x) = x^2 - 1. If gof(x) = fog(x), what is the value of a.

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

gof(x) = g(f(x)) = g(2x + a) = (2x + a)^2 - 1

fog(x) = f(g(x)) = f(x^2 - 1) = 2(x^2 - 1) + a

As gof(x) = fog(x)

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

=> 4x^2 + a^2 + 4ax - 1 = 2x^2 - 2 + a

=> 2x^2 + 4ax + a^2 - a + 1 = 0

Consider x = 0

=> a^2 - a + 1 = 0

a = `(1 - sqrt 3 i)/2` and a = `(1 + sqrt 3 i)/2`

For x = 1

8 + 4a + a^2 - a + 1 = 0

=> a^2 + 3a + 9 = 0

a = -(3*sqrt 3*i +3)/2 and a = (3*sqrt 3*i - 3)/2

It can be seen that it is not possible to provide a single value of a as for different values of x, the value of a also differs.

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