Problem 12 - College Algebra

Question

Find the domain of each function. $$ g(x)-\frac{1}{x^{2}+4}-\frac{1}{x^{2}-4} $$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \frac{1}{x^{2}+4} - \frac{1}{x^{2}-4}\) is all real numbers except \(x = 2\) and \(x = -2\).

Step 1: Identifying Denominators

The function has two denominators: \(x^{2}+4\) and \(x^{2}-4\). The function will be undefined when either of these denominators equals zero.

Step 2: Solve for when each denominator equals to zero

To find these x-values, set each denominator equal to zero and solve for x: \(x^{2}+4 = 0\) has no solutions as the equation implies that x would be an imaginary number. For \(x^{2}-4 = 0\), to solve, add 4 to each side to get \(x^{2} = 4\), then take the square root of each side to get \(x = 2\) and \(x = -2\). Thus, for \(x^{2}-4\) the function will be undefined at \(x = 2\) and \(x = -2\)

Step 3: Define the domain of the function

The domain of function g is all real numbers except for those for which the function is undefined. In this case, \(x = 2\) and \(x = -2\). Therefore, the domain of the function \(g(x) = \frac{1}{x^{2}+4} - \frac{1}{x^{2}-4}\) is \(x \neq -2, 2\).

Key Concepts

Rational Functions

Rational functions are mathematical expressions representing the ratio of two polynomials. They take a form such as \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).Understanding rational functions involves examining the behavior and restrictions that arise. A key aspect is identifying values that make the denominator zero, as these values create interruptions in the function, known as undefined points. Rational functions often involve calculations such as factoring polynomials, finding the domain, and analyzing discontinuities.In the function \( g(x) = \frac{1}{x^{2}+4} - \frac{1}{x^{2}-4} \), you can clearly see that it is composed of two separate rational expressions. Each expression is treated independently, but they collectively impact the function's domain and undefined points.

Undefined Points

Undefined points occur in a rational function when the denominator equals zero. Since division by zero is undefined in mathematics, these points need special attention to determine the function's domain.To find undefined points in \( g(x) = \frac{1}{x^{2}+4} - \frac{1}{x^{2}-4} \), set each denominator independently to zero:

• For \( x^{2} + 4 = 0 \): This equation does not yield real solutions because the sum of a square and a positive number is never zero in the real number system. Therefore, no undefined point arises from this part.
• For \( x^{2} - 4 = 0 \): Solve by rewriting it as \( x^{2} = 4 \), leading to the solutions \( x = 2 \) and \( x = -2 \). These are the values where the rational function is undefined.

The domain of a function excludes undefined points, meaning for \( g(x) \), \( x = 2 \) and \( x = -2 \) are absent from the domain.

Quadratic Equations

Quadratic equations form the basis of many mathematical calculations and have the standard form \( ax^2 + bx + c = 0 \). Solving these involves finding the values of \( x \) that satisfy the equation, often through factoring, completing the square, or using the quadratic formula.In the context of \( g(x) = \frac{1}{x^{2}+4} - \frac{1}{x^{2}-4} \), quadratic equations appear in the denominators. For instance:

• \( x^{2} + 4 = 0 \) does not produce any real roots. Its complex solutions relate to understanding when the expression equates to zero.
• In contrast, \( x^{2} - 4 = 0 \) simplifies to a solvable equation \( (x - 2)(x + 2) = 0 \), yielding roots \( x = 2 \) and \( x = -2 \). Solving these equations helps identify undefined points in the rational function.

Grasping quadratic equations is essential for managing undefined points and thus shaping the domain of rational functions.