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Fractional functions are mathematical expressions that involve a fraction. Fractions have a numerator (the top part) and a denominator (the bottom part). An important rule to remember is that the denominator of a fraction cannot be zero. Why? Because division by zero is undefined in mathematics. This is why determining where a fractional function is defined or not is crucial when working with them.

For example, in the function given, \( f(x) = \frac{2x - 1}{9 - 2x} \), the numerator \( 2x - 1 \) can be any real number. However, the denominator \( 9 - 2x \) should never be zero.

• Numerator: Can be any value.
• Denominator: Must not be zero.

By understanding fractional functions, you can prevent errors that result from undefined expressions.

An undefined expression occurs when a mathematical operation doesn't have a value. The most common cause is division by zero. This is vital to identify when analyzing functions, especially fractional functions.

For instance, if you take the function \( f(x) = \frac{2x - 1}{9 - 2x} \), you see that it becomes undefined when the denominator equals zero. To find these points:

• Set the denominator \( 9 - 2x \) to zero.
• Solve the equation \( 9 - 2x = 0 \).
• Find the value of \( x \), which is \( x = \frac{9}{2} \).

This means that \( x = \frac{9}{2} \) makes the function undefined. So, \( x \) cannot be this value when determining the function's domain.

Interval notation is a concise way to represent the domain or range of a function. It uses brackets and parentheses to show which numbers are included or excluded.

In the expression \( f(x) = \frac{2x - 1}{9 - 2x} \), we've found that the function is not defined at \( x = \frac{9}{2} \). So, we express the domain \( – the set of all x-values it can accept \) using interval notation.

For this function:

• Write all numbers from negative infinity to \( \frac{9}{2} \) \( (-\infty, \frac{9}{2}) \).
• Write all numbers from \( \frac{9}{2} \) to infinity \( (\frac{9}{2}, \infty) \).
• Combine these parts with a union symbol \( \cup \) to show the domain: \(( -\infty, \frac{9}{2} ) \cup ( \frac{9}{2}, \infty )\)

Interval notation is useful for clearly and efficiently noting where a function is defined.