In Exercises 19–32, find the (a) domain and (b) range.


𝔂 = 2 + 3x² .
x² + 4

Verified step by step guidance

1. Step 1: Identify the function given, which is \( y = \frac{2 + 3x^2}{x^2 + 4} \). This is a rational function, where the numerator is \( 2 + 3x^2 \) and the denominator is \( x^2 + 4 \).
2. Step 2: Determine the domain of the function. For rational functions, the domain is all real numbers except where the denominator is zero. Since \( x^2 + 4 \) is always positive for all real numbers (as \( x^2 \) is non-negative and 4 is positive), the denominator is never zero. Therefore, the domain is all real numbers, \( (-\infty, \infty) \).
3. Step 3: To find the range, analyze the behavior of the function as \( x \) approaches positive and negative infinity. As \( x \to \infty \) or \( x \to -\infty \), the dominant terms in the numerator and denominator are \( 3x^2 \) and \( x^2 \), respectively. Thus, the function approaches \( \frac{3x^2}{x^2} = 3 \).
4. Step 4: Consider the horizontal asymptote. Since the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients, which is \( y = 3 \). This suggests that the function approaches 3 but never actually reaches it.
5. Step 5: Evaluate the function at specific points to understand its behavior further. For example, calculate \( y \) at \( x = 0 \) to find \( y = \frac{2 + 3(0)^2}{0^2 + 4} = \frac{2}{4} = \frac{1}{2} \). This helps confirm that the range includes values below 3. Therefore, the range is \( (-\infty, 3) \).

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, like y = (2 + 3x²) / (x² + 4), the domain excludes values that make the denominator zero. In this case, since x² + 4 is always positive, the domain is all real numbers.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range, analyze the behavior of the function as x approaches various values, including infinity. For y = (2 + 3x²) / (x² + 4), consider the limits and behavior of the function to determine the range.

Behavior of Rational Functions

Understanding the behavior of rational functions involves analyzing how the function behaves as x approaches infinity or negative infinity, and identifying any asymptotes. For y = (2 + 3x²) / (x² + 4), as x becomes very large, the function approaches a horizontal asymptote, which helps in determining the range.