How to Find the Domain and Range of a Rational Function

To find the domain and range of a rational function, you need to follow a few steps.

Step 1: Determine the Domain

The domain of a rational function is all the values of x for which the function is defined. Since a rational function is defined as the ratio of two polynomials, the main concern is when the denominator is equal to zero. Start by setting the denominator equal to zero and solve for x. These x values are not included in the domain.

Example: For the function f(x) = (2x + 3) / (x – 1), set the denominator equal to zero:

x - 1 = 0  -->  x = 1

Therefore, the domain is all real numbers except x = 1, which can be written as:

Domain: x ∈ ℝ, x ≠ 1

Step 2: Determine the Range

The range of a rational function is a bit trickier to find. To determine the range, consider the horizontal asymptotes, which occur as x approaches infinity or negative infinity, and any values that f(x) cannot take.

An effective method is to analyze the leading coefficients of the numerator and denominator. If the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is less than that of the denominator, the horizontal asymptote is y = 0. If it’s greater, there is no horizontal asymptote.

In our example, since both the numerator and denominator are of degree 1, the horizontal asymptote is:

y = 2 (the leading coefficient of the numerator) / 1 (the leading coefficient of the denominator)

The function f(x) can approach this value but never actually reaches it, so y = 2 is not included in the range.

Considering all this, the range can be expressed as:

Range: y ∈ ℝ, y ≠ 2

In Summary:

• Domain: All real numbers except for values that make the denominator zero.
• Range: Values of y excluding horizontal asymptotes or any restrictions from the function itself.