For linear, quadratic, and general polynomial functions (without fractions or square roots), the domain is all real numbers, represented as (-infinity, +infinity) in interval notation.

When dealing with rational functions (fractions), the denominator cannot be zero. Set the denominator not equal to zero and solve for x. The domain excludes these x-values. For example, in 5/(x-2), x cannot be 2, so the domain is (-infinity, 2) U (2, +infinity).

For a rational function like (3x-8)/(x^2 - 9x + 20), factor the denominator to find the values of x that make it zero. In this case, x cannot be 4 or 5. The domain is (-infinity, 4) U (4, 5) U (5, +infinity).

If the denominator, when set to zero, results in an impossible condition (e.g., x^2 = -4), then there are no restrictions on the domain from the denominator. The domain is all real numbers, (-infinity, +infinity).

When the square root contains a quadratic expression (e.g., sqrt(x^2 + 3x - 28)), set the expression greater than or equal to zero. Factor the quadratic and use a sign test on a number line to determine the intervals where the expression is non-negative. For this example, the domain is (-infinity, -7] U [4, +infinity).

If the square root is in the denominator (e.g., 1/sqrt(x+3)), the expression inside the radical must be strictly greater than zero (cannot be zero). So, x+3 > 0, meaning x > -3. The domain is (-3, +infinity).

If there's a square root in the numerator and a polynomial in the denominator (e.g., sqrt(x-4)/(x^2-25)), the numerator's expression must be >= 0 (x >= 4), and the denominator cannot be zero (x != 5, x != -5). Combine these conditions on a number line. The domain is [4, 5) U (5, +infinity).

For functions like sqrt(x+3)/sqrt(x^2-16), analyze the numerator and denominator separately. The numerator, x+3, must be >= 0, so x >= -3. The denominator, x^2-16, must be > 0. Factor and use a sign test (x < -4 or x > 4). Combine these conditions by finding the intersection on a hybrid number line. The domain is (4, +infinity).

For square root functions or any radical with an even index, the expression inside the radical must be greater than or equal to zero. For example, for sqrt(x-4), x-4 >= 0, so x >= 4. The domain is [4, +infinity).