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Rational functions are a type of function that can be expressed as a fraction of two polynomials. Specifically, a rational function is written in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).

These functions can describe a variety of real-world scenarios due to their versatile nature. However, because they involve division by \( Q(x) \), you must be careful with their domains — the set of all possible input values \( x \) that make the function defined. If the denominator \( Q(x) \) is equal to zero for any value of \( x \), the function becomes undefined at that point.

Just like in the example \( h(t) = \frac{4}{t} \), we need to ensure the denominator \( t \) does not equal zero, or the function becomes undefined. This emphasizes the importance of understanding rational functions when working with division in algebra.

An expression becomes undefined when it involves division by zero. In mathematics, dividing by zero is not allowed because it doesn't result in a finite or meaningful number.

Consider an expression like \( \frac{a}{b} \). If \( b \) equals zero, the expression becomes undefined. This is because no number multiplied by zero can produce a non-zero \( a \). Attempting to divide by zero is like trying to reverse a multiplication that never made sense to begin with.

In rational functions, undefined expressions occur when the denominator is zero. For any function, finding when this happens is crucial to determining the domain. As seen in \( h(t) = \frac{4}{t} \), the expression becomes undefined when \( t=0 \). So, it's essential to exclude this value from the domain to ensure the function operates properly.

The real numbers include all the numbers you can think of that lie on the number line — these are all the numbers that are not imaginary. Real numbers encompass a wide range of values:

• All integers, both positive and negative, like -3, 0, and 4.
• All fractional numbers, like \( \frac{1}{2} \), 0.75, and 3.6.
• Irrational numbers such as \( \sqrt{2} \) and \( \pi \).

When determining the domain of a rational function, as in \( h(t)=\frac{4}{t} \), we start with the set of all real numbers. Then, we exclude those numbers that make any part of the function undefined.

In our example, the only real number that causes an issue is 0 — dividing by zero is undefined. Therefore, the domain includes all real numbers except zero. In set notation, this would be written as \( \{ t \in \mathbb{R} | t eq 0 \} \), meaning all real numbers \( t \) except for zero.