For the function f(x) = ( x2 - 4)/(x - 2), determine the domain and range.

The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function. The range of a function is the possible output values (often the "y" variable), which result from using the domain.

Step 1: Determine the Domain

The denominator of the function f(x) = (x^2 - 4)/(x - 2) cannot be zero because division by zero is undefined in mathematics. Therefore, we set the denominator equal to zero and solve for x:

x - 2 = 0 x = 2

So, the domain of the function is all real numbers except x = 2. In interval notation, this is (-∞, 2) U (2, ∞).

Step 2: Determine the Range

To find the range, we can simplify the function:

f(x) = (x^2 - 4)/(x - 2) = [(x - 2)(x + 2)]/(x - 2)

The (x - 2) terms cancel out, leaving:

f(x) = x + 2

This is a linear function, which means the range is all real numbers. However, remember that x ≠ 2, so when x = 2, there is a hole in the graph. This means the function is not defined at x = 2, so the range is all real numbers except for f(2) = 2 + 2 = 4.

So, the range of the function is all real numbers except y = 4. In interval notation, this is (-∞, 4) U (4, ∞).