Use the continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.


$g\left(x\right)=\mid \frac{x+4}{x^2-4}\mid$

Verified step by step guidance

1. Identify the function: \( g(x) = \left| \frac{x+4}{x^2-4} \right| \).
2. Recognize that the absolute value function is continuous everywhere, so focus on the rational function \( \frac{x+4}{x^2-4} \).
3. Determine where the denominator \( x^2 - 4 \) is zero, as these points will be discontinuities. Solve \( x^2 - 4 = 0 \) to find \( x = 2 \) and \( x = -2 \).
4. Conclude that the function \( g(x) \) is continuous on the intervals where the denominator is not zero: \((-\infty, -2) \cup (-2, 2) \cup (2, \infty)\).
5. Verify that the function is not defined at \( x = 2 \) and \( x = -2 \), confirming these are points of discontinuity.

Key Concepts

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. Understanding continuity is essential for determining where the given function behaves without interruption.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x, effectively removing any negative sign. This function is continuous everywhere on the real number line. When analyzing functions involving absolute values, it is crucial to consider how the absolute value affects the overall continuity and behavior of the function.

Rational Functions and Their Discontinuities

A rational function is a ratio of two polynomials. Discontinuities in rational functions typically occur where the denominator equals zero, leading to undefined values. In the case of the function g(x) = |(x + 4)/(x^2 - 4)|, identifying the points where the denominator (x^2 - 4) is zero is vital for determining the intervals of continuity.