Problem 18 - Algebra A Combined Function

Question

Find any numbers for which each rational expression is undefined. $$ \frac{9 y^{5}+y^{3}}{9} $$

Step-by-Step Solution

Answer

The expression is defined for all real numbers.

Step 1: Identify the Denominator

To determine where the rational expression is undefined, identify the denominator of the fraction. In this expression, the denominator is 9.

Step 2: Determine Denominator Non-Zero Values

The expression will be undefined for any value that makes the denominator equal to zero. Since the denominator is just a constant (9), it is never zero. Therefore, there are no values that make the denominator zero.

Step 3: Conclude Domain for Undefined Points

Since 9 is a constant and does not contain a variable, there are no values of the variable that make the denominator zero. Thus, the expression is defined for all real numbers.

Key Concepts

Undefined Expressions

An undefined expression in mathematics is one in which the calculations do not yield a finite or meaningful result. In the case of a rational expression, undefined points typically occur when the denominator is equal to zero. This is because division by zero is not possible in mathematics. To determine when a rational expression is undefined, you need to set the denominator equal to zero and solve for the variable. In our example, the denominator is always 9, which is a non-zero constant. Therefore, there are no values of the variable that make the denominator zero. So, this rational expression has no undefined points and is valid for all real numbers.

Denominator

The denominator is the bottom part of a fraction. It tells us into how many parts the whole is divided. In rational expressions, it's vital to pay attention to the denominator to avoid division by zero, which would make the expression undefined.In the provided example, the expression is \(\frac{9y^5 + y^3}{9}\). Here, the denominator is a constant number 9. Since 9 is a positive, non-zero real number, the denominator does not affect the definition of the expression negatively. In more complex rational expressions, if the denominator includes variables, it is necessary to find the variable values that make the denominator zero. This helps identify where the expression may become undefined.

Real Numbers

Real numbers include all the numbers on the number line. This set includes rational numbers (like fractions and terminating decimals), irrational numbers (like \(\pi\) or non-terminating, non-repeating decimals), integers, zero, and even negative numbers. In the context of rational expressions, when the denominator remains non-zero and constant, as with the number 9 in our example, the expression is defined for all real numbers. This means regardless of what value \(y\) takes, the expression \(\frac{9y^5 + y^3}{9}\) remains valid. The lack of restriction on the variable\(y\) makes the domain of the expression inclusive of all real numbers.