91Ó°ÊÓ

Rational functions are mathematical expressions that represent the ratio of two polynomials, essentially a fraction in which both the numerator and the denominator are polynomials. The general form of a rational function is given by
\( f(x) = \frac{p(x)}{q(x)} \)
where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) \) is not the zero polynomial. The beauty of rational functions lies in their ability to model various real-world situations, such as rates of change, and they hold a significant place in subjects like mathematics, physics, and engineering.
An essential aspect of rational functions involves their domains. Since division by zero is undefined, any value of \( x \) that causes the denominator \( q(x) \) to be zero must be excluded from the domain. The domain of a rational function, therefore, consists of all real numbers except for the zeros of the denominator polynomial.

Understanding the domain of a function is crucial since it informs us of all the possible input values (or \( x \) values) that the function can accept. Establishing the domain ensures that the function operates within the limits of mathematical principles, such as avoiding division by zero or the extraction of square roots from negative numbers in real number contexts.
To determine the domain of a function algebraically, one needs to:

• Identify any potential restrictions such as divisions by zero, square roots of negative numbers, or logarithms of non-positive numbers.
• Set constraints that prevent these undefined behaviors.
• Solve the constraints to find the values that must be excluded from the domain.

For polynomial and rational functions, the process typically focuses on identifying the values that force the denominator to zero. For functions involving roots, attention is on ensuring the radicand is non-negative. By systematically applying these principles, one can accurately determine the function's domain.

Finding the domain of a function algebraically requires a step-by-step approach that takes into account the specific type of function we are dealing with. For example, when dealing with rational functions, we first need to set the denominator equal to zero and solve for any values of the variable that satisfy this condition. These values are then excluded from the domain.
Let's consider the function \( s(y) = \frac{3y}{y+5} \). Following the steps for determining the domain algebraically, we identify that the denominator is \( y+5 \) and it cannot be zero. Therefore, setting \( y+5 = 0 \) and solving for \( y \) gives us \( y = -5 \) which is the value that must be excluded from the domain. Consequently, the domain of \( s(y) \) is all real numbers except for \( y = -5 \) because including \( y = -5 \) would make the denominator zero, leading to an undefined function value.