91Ó°ÊÓ

Understanding the zero of a denominator is essential when working with rational functions. A rational function is a fraction where both the numerator and denominator are polynomial expressions. The denominator cannot be zero since division by zero is undefined in mathematics. Hence, to find the problematic points in the domain, you must set the denominator equal to zero and solve for the variable.
For example, in the function given by the exercise, the denominator is \((x-5)(x+4)\). To find when this fraction is undefined, you solve \((x-5)(x+4) = 0\). This will give the values of \(x\) that make the denominator zero. These are the points where the rational function does not exist.
By solving, you find that \(x = 5\) and \(x = -4\) are the zeros of the denominator. These values of \(x\) need to be excluded from the domain of the function.

Factoring equations is an efficient way to solve for unknown values, particularly when dealing with polynomials. Factored equations break down complex expressions into simpler products. This method is especially useful in finding critical points in rational functions.
When you have a factored equation like \((x-5)(x+4) = 0\), each factor can be set to zero to solve for \(x\). This step is crucial in identifying the zeros of the denominator.
For instance:

• Set \(x-5 = 0\): Solve for \(x\) to get \(x = 5\).
• Set \(x+4 = 0\): Solve for \(x\) to get \(x = -4\).

The solutions represent the values of \(x\) which would make the denominator zero, thereby helping to determine points to exclude from the domain of the function.

The domain of a function is the set of all potential input values. In the context of rational functions, the domain represents all real numbers except those that make the denominator zero. Excluding these values ensures you only include points where the function is defined.
To find the domain of \(g(x)=\frac{3x^{2}}{(x-5)(x+4)}\), we exclude \(x = 5\) and \(x = -4\), the points where the denominator is zero.
This creates intervals over which the function is defined. Thus, the domain can be expressed in interval notation as:

• \((-\infty, -4)\)
• \((-4, 5)\)
• \((5, +\infty)\)

These intervals represent all the real numbers that can be plugged into the function without causing the denominator to become zero, giving a complete picture of the rational function's domain.