Problem 31 - Calculus with Concepts in Calculus

Question

Find the domain of the function. $$ k(x)=\frac{2 x-3}{x^{2}+4} $$

Step-by-Step Solution

Answer

The domain of the function is all real numbers.

Step 1: Understand the Function

The function we are examining is \( k(x) = \frac{2x - 3}{x^2 + 4} \). This function is a rational function, meaning it is formed by dividing one polynomial by another. The domain of a rational function is typically all real numbers except where the denominator is zero.

Step 2: Analyze the Denominator

For the domain of \( k(x) \), we need to determine when the denominator, \( x^2 + 4 \), is equal to zero. We solve the equation \( x^2 + 4 = 0 \).

Step 3: Solve for the Denominator

Set \( x^2 + 4 = 0 \) and solve for \( x \):\[ x^2 + 4 = 0 \]This simplifies to \( x^2 = -4 \). Because \( x^2 = -4 \) has no real solutions (since the square of any real number is non-negative), the denominator cannot be zero for any real \( x \).

Step 4: Determine the Domain

Since the denominator \( x^2 + 4 \) is never zero for any real \( x \), all real numbers are within the domain of \( k(x) \). Thus, there are no restrictions for \( x \) and the domain of the function is all real numbers.

Key Concepts

Rational Function

A rational function represents a function that is the quotient of two polynomials. In simpler terms, it is a fraction where the numerator and the denominator are both polynomials. For example, in the function \( k(x) = \frac{2x - 3}{x^2 + 4} \), "2x - 3" is a polynomial in the numerator, and "x^2 + 4" is a polynomial in the denominator. Rational functions are intriguing because they can represent a wide range of curves. However, the mere presence of polynomials in a fraction also leads to specific considerations like undefined points where the denominator is zero. Understanding rational functions is a crucial aspect of algebra, paving the way for deeper insights in calculus.Additionally, rational functions are often used to model real-world scenarios, such as supply and demand curves in economics or concentration gradients in chemistry. Grasping how these functions behave can offer valuable tools for analysis across multiple disciplines.

Denominator Analysis

To find the domain of a rational function, one of the key steps is to perform a denominator analysis. This involves identifying the values for which the denominator equals zero, as these are the points where the function becomes undefined.In our example, the denominator is \( x^2 + 4 \). We set up the equation \( x^2 + 4 = 0 \) to determine the problematic points:

• Rearrange to \( x^2 = -4 \).
• A critical observation here: for real numbers, there cannot be a solution, since the square of a real number can't be negative.

This means that, for real numbers, there is no value of \( x \) that makes \( x^2 + 4 = 0 \). This insight shows us that our rational function \( k(x) = \frac{2x - 3}{x^2 + 4} \) has no restrictions based on the denominator for real values of \( x \).

Real Numbers Domain

When discussing the domain of functions, particularly rational functions, it usually involves discussing all real numbers minus the points that make the function undefined. In the case of rational functions, this comes down to when the denominator is zero.For the function \( k(x) = \frac{2x - 3}{x^2 + 4} \), we examined the denominator \( x^2 + 4 \). Since we've established that this expression is never zero for any real value of \( x \), the domain of \( k(x) \) includes all real numbers.This means that any real number can be popped into \( k(x) \) without causing it to become undefined. Such an outcome is relatively rare. Most rational functions will have some values omitted in their domain. However, with \( k(x) \), the overall result is a significant simplification, highlighting the function's unrestricted nature across the real number line. When determining the domain of any function, always assess the denominator carefully to capture all nuances.