91Ó°ÊÓ

Rational functions are an important topic in algebra. A rational function is a fraction where both the numerator and the denominator are polynomials. For example, the function given in the exercise, \( g(x)=\frac{x-2}{x+4} \), is a rational function. Here, \( x-2 \) is the numerator and \( x+4 \) is the denominator. Rational functions can often look complex, but solving problems involving them usually involves straightforward steps. Understanding the behavior of these functions helps in solving a variety of equations.

When solving equations, particularly rational functions, the key is to understand how to find solutions that make the entire function equal to a given value. In the exercise, we needed to solve \( g(x)=0 \). The steps were:

• Set the function equal to zero: \( \frac{x-2}{x+4} = 0 \).
• Recognize that a fraction is zero when its numerator is zero, which leads us to \( x-2 = 0 \).
• Solve for \( x \) to find \( x=2 \).
• Check that this solution does not make the denominator zero by substituting back into the denominator \( x+4 \) and confirming it is non-zero.

These are typical steps for solving rational equations, helping us understand the values where the function meets specific criteria.

A clear understanding of numerators and denominators is essential in working with rational functions. In the function \( g(x)=\frac{x-2}{x+4} \), \( x-2 \) is called the numerator, and \( x+4 \) is called the denominator. The numerator decides when a rational function is zero. This is because a fraction equals zero when its numerator is zero, as long as the denominator is not zero. For the given function, we set \( x-2 = 0 \) and solved for \( x \). The denominator, on the other hand, cannot be zero because division by zero is undefined. Thus, when verifying solutions, we must ensure the denominator remains non-zero. This dual focus on both the numerator and denominator is crucial in understanding and solving rational functions.