91Ó°ÊÓ

A rational function is any function that can be expressed as the quotient of two polynomials. In other words, it is a fraction where both the numerator and the denominator are polynomials.
For example, the function \( f(u) = \frac{2u + 7}{\sqrt{u+5}} \) can be seen as a rational function if we consider the denominator as the expression involving the square root as part of a polynomial equation.

• Numerator: \(2u + 7\)
• Denominator: \(\sqrt{u+5}\)

The behavior of rational functions is largely dictated by their denominators. This is because a function becomes undefined when its denominator is equal to zero. These are often the points where the function is discontinuous, as seen in the given problem where the denominator is zero at \(u = -5\). Therefore, understanding the denominator is key to understanding where the function might have discontinuities or breaks.

Square root functions involve expressions with the square root of a variable. These functions are often represented as \( f(x) = \sqrt{x} \), where \(x\) must be non-negative for the square root to be defined in the real number system.
In the expression \( \sqrt{u+5} \), inside the denominator of our rational function, the term under the square root must always be greater than or equal to zero. This means that \( u+5 \geq 0 \), or equivalently, \( u \geq -5 \).

• Expression: \( \sqrt{u+5}\)
• Condition: \( u+5 \geq 0\)

This requirement ensures that the square root does not attempt to evaluate a negative number, which would be undefined in real numbers. This highlights an important aspect of understanding square root functions: always consider the domain, or the set of allowable inputs.

The domain of a function consists of all possible input values (usually represented as \(x\) or \(u\)) for which the function is defined. For our function \( f(u) = \frac{2u + 7}{\sqrt{u+5}} \), the domain is determined by the constraints that make both the numerator and denominator valid.
Particularly for this function, you have two considerations:

• The denominator cannot be zero.
• The expression under the square root must be non-negative.

This means:

• \( \sqrt{u+5} eq 0 \rightarrow u eq -5\).
• \( u+5 \geq 0 \rightarrow u \geq -5\).

Thus, the domain of \( f(u) \) is \( u > -5 \). The function is not defined for \( u \leq -5 \), confirming that it is discontinuous at \( u = -5 \), as the function jumps from being undefined to defined just after this point.