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A rational function is defined as a fraction where both the numerator and the denominator are polynomials. When we look at a function like
\( s(y) = \frac{3y}{y+5} \),
we see a classic example of a rational function. The numerator is the polynomial
\( 3y \),
and the denominator is
\( y + 5 \).

When learning about rational functions, it's crucial to understand their properties, like the domain, which tells us all the possible input values for the function. For rational functions, the domain typically includes all real numbers, except those that make the denominator equal zero because division by zero is undefined.

A good exercise improvement advice for students struggling with rational functions is to plot the function on a graph. This visual representation helps illustrate how the function behaves and especially where it is not defined (in this case at
\( y = -5 \)).

Within the study of functions, the concept of undefined function values can often cause confusion. A function value becomes undefined when the function involves some form of invalid arithmetic operation, such as division by zero. As highlighted in the previous section, in rational functions, when the denominator is
\( y+5 \),
the function becomes undefined at the point where
\( y+5 = 0 \).

To grasp this concept fully, one must remember that any number divided by zero does not result in infinity or a very large number, as sometimes mistakenly believed. It leads to a complete break in the mathematical logic we rely on, thus 'undefined'. A good way to remember this is to think about dividing something into zero groups, which doesn't make sense: you simply cannot have zero groups of something.

The point where the function is undefined is not a hole in the graph; it's often referred to as a vertical asymptote— a line that the graph approaches but never touches.

Solving equations is a fundamental concept in algebra that involves finding the values for which an equation holds true. The process often entails a series of steps where we manipulate the equation by performing operations such as adding, subtracting, multiplying, or dividing both sides to isolate the unknown variable.

For instance, in the exercise, we have the equation
\( y+5 = 0 \).
The solution involves a basic algebraic step: subtracting 5 from both sides. This yields
\( y = -5 \),
which is the value making our original function undefined.

When solving equations, especially those that inform the domain of a function, it's essential to check our solutions to ensure they make sense within the context of the given function. In this case, students can improve their understanding by re-inserting the solution back into the function to verify that it does indeed produce an undefined value.