Problem 13 - The Calculus with Analytic Geometry

Question

Determine the intervals on which the given function is continuous. $$ f(x)=\frac{1}{x^{2}-4} $$

Step-by-Step Solution

Answer

The function is continuous on \((-\infty, -2)\) \(U (-2, 2)\) \(U (2, \infty)\).

Step 1: Identify the function's domain

The function is given as \(f(x) = \frac{1}{x^2 - 4}\). This function is a rational function, which means it is continuous where its denominator is not zero.

Step 2: Set the denominator equal to zero

To find where the function is not continuous, set the denominator equal to zero: \(x^2 - 4 = 0\).

Step 3: Solve for the roots

Solve the equation \(x^2 - 4 = 0\):\[ x^2 = 4 \]\[ x = \pm 2 \].

Step 4: Determine the intervals of continuity

The function is not continuous at \(x = 2\) and \(x = -2\). Thus, the function is continuous on all intervals except at these points. The intervals of continuity are:\((-\infty, -2)\) \(U (-2, 2)\) \(U (2, \infty)\).

Key Concepts

rational function

A rational function is a fraction where both the numerator and the denominator are polynomials. In mathematical terms, it takes the form \( \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials. Unlike regular polynomials, rational functions have restrictions for their domain due to the denominator.

If the denominator of a rational function is zero, the function will not be defined at that point. This leads to discontinuities, which are important for understanding the behavior of the function.

domain of a function

The domain of a function consists of all possible input values (x-values) for which the function is defined. For rational functions, checking the domain is essential to avoid division by zero.

For the given function \(f(x) = \frac{1}{x^2 - 4} \), the first step is identifying where the denominator is zero. This means solving the equation Let’s set \( x^2 - 4 = 0\).

Solving the equation, we get:
\[ x^2 = 4 \]
\[ x = \pm 2 \]

Thus, \( x = 2 \) and \( x = -2 \) are points where the function is not defined. Therefore, the domain includes all real numbers except these points.

continuity intervals

Continuity of a function means there are no breaks, jumps, or holes within its graph. For rational functions, the function is generally continuous except where the denominator is zero.

For \( f(x) = \frac{1}{x^2 - 4} \), we need to find these intervals of continuity. We've previously identified that \( x = 2\) and \( x = -2 \) make the denominator zero, making the function discontinuous there.

So, to define the intervals of continuity, split the number line at these points:

• \((- \infty , -2 ) \)
• \(( -2, 2 ) \)
• \(( 2, \infty ) \)

The function is continuous within these intervals but not at \( x = 2 \) and \( x = -2 \).

solving equations

Solving equations is fundamental in finding critical points like where a function might be undefined. To find where \( f(x) = \frac{1}{x^2 - 4} \) is undefined, we solved \( x^2 - 4 = 0 \).

Here's how this equation is solved step-by-step:

1. Start with \( x^2 - 4 = 0 \).
2. Add 4 to each side to get \( x^2 = 4 \).
3. Take the square root of both sides, leading to \( x = \pm 2 \).

This process helps us understand why \( x = 2 \) and \( x = -2 \) are critical points affecting the function’s continuity. Functions often involve solving such equations to understand their properties and behavior.