Determine f(x) if f(x)+3f(1-x)=x^2+4x+7

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justaguide | College Teacher | (Level 2) Distinguished Educator

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It is given that: f(x) + 3*f(1 - x) = x^2 + 4x + 7

Let f(x) = ax^2 + bx + c

f(1 - x) = a(1 - x)^2 + b(1 - x) + c = a + ax^2 - 2ax  + b - bx + c = ax^2 - x(2a+b)+ a + b + c

f(x) + 3*f(1 - x) = ax^2 + bx + c + 3ax^2 - x*3(2a+b) + 3(a+b+c)

ax^2 + bx + c + 3ax^2 - x*3(2a+b) + 3(a+b+c) = x^2 + 4x + 7

=> x^2(a+3a) + x(b-6a-3b) + c + 3a +3b+3c = x^2 + 4x + 7

=> x^2(4a) + x(-6a-2b) + 3a+3b+4c = x^2 + 4x + 7

=> 4a = 1, -6a-2b = 4 and 3a+3b+4c = 7

=> a = 1/4

substitute a= 1/4 in -6a-2b = 4

=> -6/4 - 4 = 2b

=> b = -3/4 - 2

=> b = -11/4

substitute a = 1/4 and b = -11/4 in 3a+3b+4c = 7

=> 3/4 - 33/4 + 4c = 7

=> c = 29/8

The function f(x) = x^2/4 - 11x/4 + 29/8

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