Verified Answer :

  • Determine the domain by setting the denominator of the rational function to zero: \(x^2 - 16 = 0\).
  • Factor the expression to solve for the excluded values: \((x - 4)(x + 4) = 0\), which gives \(x = 4\) and \(x = -4\).
  • Exclude these values from the real numbers to define the domain.
  • The final domain is \((-\infty, -4) \cup (-4, 4) \cup (4, \infty)\) or \(\boxed{x \in \mathbb{R}, x eq \pm 4}\).

Explanation

  1. Identifying the Condition for the Domain

To find the domain of the function \(g(x) = \frac{3x}{x^2 - 16}\), we identify the values of \(x\) for which the function is defined. A rational function like this one is defined everywhere except where its denominator is equal to zero. Therefore, we must find the values of \(x\) that satisfy the equation \(x^2 - 16 = 0\).

  1. Solving for Forbidden Values

We solve the equation \(x^2 - 16 = 0\) by recognizing it as a difference of squares. We can factor this expression:
\[x^2 - 16 = (x - 4)(x + 4) = 0.\]
Setting each factor equal to zero, we find the roots:
\[x - 4 = 0 \implies x = 4,\]
\[x + 4 = 0 \implies x = -4.\]
Thus, the denominator is zero at \(x = 4\) and \(x = -4\).

  1. Stating the Domain

Since the function is undefined at \(x = 4\) and \(x = -4\), we exclude these points from the set of all real numbers. The domain of the function is the set of all real numbers \(x\) such that \(x eq 4\) and \(x eq -4\). In interval notation, this is represented as \((-\infty, -4) \cup (-4, 4) \cup (4, \infty)\).

Examples

In real-world applications, identifying the domain of a function is crucial when modeling scenarios like signal transmission or physical constraints in engineering. For instance, if a formula describes the voltage in a circuit, values that make the denominator zero represent "short circuits" or points where the physical system cannot exist. By calculating the domain, engineers avoid these critical points to ensure the stability and safety of their systems.

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