Linear function Discover the linear function? f(x+2)*f(x-2) = x^2-2x-3

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to substitute `x + 2` and ` x - 2` for x in equation of function `f(x) = mx + n` , such that:

`f(x + 2)*f(x - 2) = (m(x + 2) + n)(m(x - 2) + n)`

`f(x + 2)*f(x - 2) = (mx + n + 2m)(mx + n - 2m)`

You may convert the product of factors into a difference of two squares, such that:

`f(x + 2)*f(x - 2) = (mx + n)^2 - (2m)^2`

`f(x + 2)*f(x - 2) = f(x)^2 - 4m^2`

The problem provides the information that `f(x + 2)*f(x - 2) = x^2 - 2x - 3` , such that:

`(mx + n)^2 - 4m^2 = x^2 - 2x - 3`

`m^2x^2 + 2mnx + n^2 - 4m^2 = x^2 - 2x - 3`

Equating the coefficients of like powers yields:

`m^2 = 1 => m = +-1`

`2mn = -2 => mn = -1 => n = +-1`

Hence, evaluating the functions that consider the given conditions, yields `f(x) = x - 1` and `f(x) = -x + 1.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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