Question

Find the domain of the function. \(s(y)=\frac{3 y}{y+5}\)

Short answer

The domain of the function \(s(y)=\frac{3 y}{y+5}\) is all real numbers except -5.

Step-by-step solution

Identifying the denominator

The denominator of the fraction in the function is \(y+5\). In order for the function to be defined, the denominator should not be equal to 0.

Setting the denominator equal to zero and solving

Set the denominator equal to zero and solve for 'y': \(y + 5 = 0\). Solving this equation gives \(y = -5\).

Excluding the undefined value from the domain

The domain of 'y' is all real numbers except the one that makes the denominator equal to 0, i.e., \(y \neq -5\). So the domain is all real numbers except -5.

Key concepts

Rational Functions

Rational functions are mathematical expressions representing the division of two polynomials. They take the general form of \( f(x) = \frac{p(x)}{q(x)} \) where both \( p(x) \) and \( q(x) \) are polynomials and \( q(x) \) is non-zero. Since dividing by zero is undefined in mathematics, we carefully examine a rational function's denominator to understand its behavior.

For instance, in the given exercise \( s(y)=\frac{3y}{y+5} \), since the numerator \( 3y \) is a simple first-degree polynomial and the denominator is \( y+5 \)—also a polynomial—we recognize this function as a rational function. The domain of this function includes all values that 'y' can take, except for those which would make the denominator equal to zero. This approach of determining the suitable values of 'y' is essential in preventing undefined expressions within rational functions.

Excluded Values

When dealing with functions, and more specifically with rational functions, excluded values have a significant role. These are the values that we must omit from the domain of a function because they would cause a division by zero, which is undetermined in the realm of real numbers. Excluding these values ensures that the function is well-defined for all inputs in its domain.

As shown in the solution to the exercise, the excluded value is the solution to the equation \( y+5=0 \), which simplifies to \( y=-5 \). Since this value would make the denominator of our function zero, it must be removed from the domain. Recognizing and excluding these values is a vital step when determining the domain of rational functions.

Real Numbers

Real numbers encompass both rational numbers (like fractions and integers) and irrational numbers (like \( \sqrt{2} \) or \( \pi \) ). In defining the domain of functions, especially rational functions, we often seek a range of real numbers for which the function can produce real outputs. This means excluding any inputs that result in non-real outputs, such as dividing by zero.

Thus, when we state that the domain of the function \( s(y) \) is all real numbers except for -5, we signify that any real number 'y' can be used in the function, provided that it does not lead to division by zero. The real numbers are the backbone of continuous domains in which functions such as \( s(y) \) operate, minus the defined exceptions, which in this case is the single excluded value \( y = -5 \).