What is the linear function?f(x+2)*f(x-2) = x^2-2x-3

We'll write 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:

f(x) = x - 1, for a = 1 and b =
-1

or

f(x)
= -x + 1, for a = -1 and b = 1.