# Linear function.Solve f(x+2)*f(x-2) = x^2-2x-3.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll write the standard form of the linear function:

f(x) = ax + b

To determine the function, we'll have to determine the coefficients a and b.

For this reason, we'll use the constraint given by enunciation:

f(x+2)*f(x-2) = x^2-2x-3

We'll write f(x+2):

f(x+2) = a(x+2) + b

f(x+2) = ax + 2a + b (1)

We'll write f(x-2):

f(x-2) = a(x-2) + b

f(x-2) = ax + b - 2a (2)

We'll multiply (1) by (2):

f(x+2)*f(x-2) = (ax + 2a + b)(ax + b - 2a)  = (ax+b)^2 - (2a)^2

We'll expand the squares:

f(x+2)*f(x-2) = a^2*x^2 + 2abx + b^2 - 4a^2 (3)

But f(x+2)*f(x-2) = x^2-2x-3 (4)

We'll put (3) = (4)

a^2*x^2 + 2abx + b^2 - 4a^2 = x^2 - 2x - 3

a^2 = 1

a = -1 or a = 1

2ab = -2

ab = -1

If a = 1 => b = -1

If a = -1 => b = 1

So, the linear function could be:

If a = 1 and b = -1, then the linear function is: f(x) = x - 1

If a = -1 and b = 1, then the linear function is: f(x) = -x + 1