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The domain of a function is important to understand. It tells us what inputs (or values of the independent variable, often represented as \(x\)) we can use in the function. For a rational function like \(g(x) = \frac{4-3x}{2}\), we need to look for situations that might make the function undefined. A rational function is undefined when its denominator equals zero. This is because division by zero is not possible. In the function \(g(x)\), the denominator is a constant number, 2. Since 2 is never zero, there are no restrictions on the values that \(x\) can take. Therefore, the domain of \(g(x)\) is all real numbers, meaning you can substitute any real number for \(x\) and calculate \(g(x)\) without issues. To put it simply, you don't need to exclude any numbers for \(x\) when the denominator doesn't involve \(x\).

Understanding the numerator and denominator is crucial when dealing with rational functions. A rational function typically has the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomial expressions. In \( g(x) = \frac{4-3x}{2} \), the numerator is \( 4 - 3x \) and the denominator is 2. The numerator can be any expression, and it determines the values that the function might produce.

• Numerator: \(4 - 3x\) gives the overall shape of the function. Changes in \(x\) modify the output \(g(x)\).
• Denominator: A constant of 2, meaning \(g(x)\) never hits any undefined point.

Whenever you're asked to find the domain of a rational function, focus on the denominator. Make sure it doesn’t equal zero. Here, as the denominator is constant and non-zero, it simplifies the domain analysis.

Real numbers are the set of numbers you can find on the number line. They include both positive and negative numbers, as well as zero. Real numbers encompass:

• Integers, like -3, 0, 1
• Fractions, like \(\frac{1}{2}\), \(-\frac{3}{4}\)
• Decimals, like 0.25, -4.5
• Irrational numbers, which cannot be expressed as a simple fraction, like \(\pi\) and \(\sqrt{2}\)

For a rational function, the domain is usually all real numbers except where the denominator equals zero. But in \(g(x) = \frac{4 - 3x}{2}\), since the denominator is constant and doesn't involve \(x\), there's no value that can make it zero. This means the function happily accepts any real number as an input for \(x\). Consequently, the domain of the function \(g(x)\) is all real numbers, meaning \((-\infty, \infty)\). This shows the diverse nature and scope of real numbers in defining domains of functions.