Question

In Exercises 19–32, find the (a) domain and (b) range.𝔂 = 2 + 3x² .x² + 4

Accepted Answer

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 $$ x^2 + 4 . $$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, $$ (-\infty, \infty) . $$Step 3: To find the range, analyze the behavior of the function as $$ x $$ approaches positive and negative infinity. As $$ x 	o \infty  or$$ $$ x 	o -\infty $$, the dominant terms in the numerator and denominator are $$ 3x^2 $$ and $$ x^2 $$, respectively. Thus, the function approaches $$ \frac{3x^2}{x^2} = 3 . $$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. 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 $$ (-\infty, 3) .$$

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

𝔂 = 2 + 3x² .
x² + 4

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Key Concepts

Domain of a Function

Range of a Function

Behavior of Rational Functions