How can I determine the domain and range of a rational function?

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Finding the Domain and Range of a Rational Function

A rational function is a function that can be expressed as the quotient of two polynomials, typically in the form:

f(x) = P(x) / Q(x)

where P(x) and Q(x) are polynomials.

Step 1: Determine the Domain

The domain of a rational function consists of all real numbers except those that make the denominator zero. To find the domain, follow these steps:

  1. Identify the denominator of the rational function, which is Q(x).
  2. Solve the equation Q(x) = 0 to find the values of x that are not included in the domain.
  3. The domain can be expressed in interval notation by omitting the values that make the denominator zero.

For example, for the rational function f(x) = (2x + 3) / (x – 1), the denominator is x – 1. Setting this equal to zero gives:

x - 1 = 0 → x = 1

This indicates that the domain is all real numbers except x = 1, which can be written in interval notation as:

( -∞, 1 ) ∪ ( 1, ∞ )

Step 2: Determine the Range

The range of a rational function is more complex, as it involves the values that the function can take. To find the range, consider the following steps:

  1. Analyze the behavior of the function as x approaches the points where the denominator is zero, including any vertical asymptotes.
  2. Check for horizontal asymptotes by finding the limit of f(x) as x approaches positive and negative infinity. The horizontal asymptote can suggest values that f(x) cannot reach.
  3. Evaluate the function at critical points and analyze the overall shape of the graph to ascertain all possible y values.

Continuing with the previous example, as x approaches 1, f(x) approaches ±∞, indicating a vertical asymptote at this point. Now, considering limits at infinity:

limx→∞ f(x) = 2,
limx→-∞ f(x) = 2

Since the degrees of the numerator and denominator are the same and the leading coefficients are equal, the horizontal asymptote is y = 2. Hence, the function never actually reaches this value.

However, f(x) can take on any other real number, leading to the conclusion that the range is:

( -∞, 2 ) ∪ ( 2, ∞ )

Final Summary

  • Domain: All real numbers except where the denominator is zero.
  • Range: All real numbers except the horizontal asymptote’s value.

Understanding the domain and range of rational functions helps in graphing and analyzing their behavior effectively. Happy learning!

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