Algebra - Domain of Factored Rational Expression?

The one to watch for is the denominator.

The reason: divisions by zero are not defined.

So, when the expression (x+2)^2 is equal to zero (that is, when x = -2), the function

f(x) = (x-2)^2 / (x+2)^2

will be undefined.

Therefore, the value -2 cannot be in the domain.

Otherwise, the function will work very well for any other value.

Domain = the union of two intervals:

(-infinity to -2) where -2 is not included

and

(-2 to +infinity) where -2 is not included.

[the infinities are never included because they are not well-defined values]

-----

The numerator is allowed to be zero, that is not a problem.

If x = +2, you'd get

f(2) = (2 - 2)^2 / (2 + 2)^2 = 0^2 / 4^2 = 0/16 = 0

and 0 is a well-defined value.

The domain of a function is the set of all values for which the function is defined. That is, the function provides an "output" or value for each member of the domain.

As you can see here the function is a quotient so be careful with the denominator cause you cannot divide by zero..

So,you demand (x-2)^2 ≠ 0

x ≠ 2

Finally the domain is all the real numbers except for 2.

Factored, you have

(x + 2)(x - 2) / (x + 2)(x + 2)

The (x + 2) in the numerator cancels with the one in the denominator, leaving you with

(x - 2) / (x + 2)

So x can be anything except -2.

The domain is the set of all real x excluded those values that make the denominator equal to 0

D = R \ {-2} or

D = (-infinity,-2) U (-2,+infinity)

(-15x^3y^3)/(-20xy^4) (-15)/(-20) = 3/4 x^3/x = x^2 y^3/y^4 = a million/y (-15x^3y^3)/(-20xy^4) = 3x^2/4y (4x - 28)/(5x - 35) = [4(x - 7)]/[5(x - 7)] = 4/5 (4r^2 - 25s^2)/(2r^2 + 3rs - 20s^2) The polynomials could decide to be factored 4r^2 - 25s^2 = (2r + 5s)(2r - 5s) {huge distinction of two squares) 2r^2 + 3rs - 20s^2) = (2r - 5s)(r + 4s) The fraction is now [(2r + 5s)(2r - 5s)]/[(2r - 5s)(r + 4s)] (2r + 5s)/(r + 4s) See the factoring lesson on the link under