Problem 10 - Algebra for College Students

Question

Find the domain of each rational expression. $$\frac{z-12}{4 z}$$

Step-by-Step Solution

Answer

The domain is all real numbers except \(z = 0\).

Step 1 - Identify the Rational Expression

The given rational expression is \(\frac{z-12}{4z}\).

Step 2 - Understand the Domain

The domain of a rational expression consists of all real numbers except where the denominator is zero.

Step 3 - Find the Denominator

In the expression \(\frac{z-12}{4z}\), the denominator is \(4z\).

Step 4 - Set Denominator Equal to Zero

Set the denominator \(4z\) equal to zero and solve for \(z\): \[4z = 0\]

Step 5 - Solve for z

Solve the equation \(4z = 0\): \[z = 0\]

Step 6 - Write the Domain

Since \(z = 0\) makes the denominator zero, the domain excludes \(z = 0\). Thus, the domain is all real numbers except \(z = 0\).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Think of these like regular fractions but with algebraic expressions instead of just numbers. For example, in the rational expression \(\frac{z-12}{4z}\), \(z-12\) is the numerator and \(4z\) is the denominator. These expressions are common in algebra and higher math.

Key points about rational expressions:

• They can be simplified just like regular fractions.
• It's important to remember that the denominator cannot be zero. If the denominator is zero, the expression is undefined.
• When working with these expressions, always factor and simplify both the numerator and the denominator where possible.

Domain of a Function

The domain of a function represents all the possible input values (typically, values of \(x\)) that the function can accept without leading to any mathematical errors. For rational expressions like \(\frac{z-12}{4z}\), these errors typically arise when the denominator equals zero because division by zero is undefined.

Steps to find the domain for rational expressions:

• Identify the denominator.
• Set the denominator equal to zero and solve for the variable.
• Exclude these values from the set of all real numbers.
• The remaining values form the domain.

In our example, the domain of \(\frac{z-12}{4z}\) excludes \(z = 0\) because it makes the denominator zero. So, the domain is all real numbers except \(z = 0\).

Denominator

The denominator is the bottom part of a fraction or rational expression. In the rational expression \(\frac{z-12}{4z}\), \(4z\) is the denominator. The fundamental rule is the denominator cannot be zero.

To determine when the denominator makes the rational expression undefined:

• Set the denominator equal to zero.
• Solve the equation.
• The solutions are the values that must be excluded from the domain.

Here, setting \(4z = 0\) and solving for \(z\) gives \(z = 0\). This tells us that \(\frac{z-12}{4z}\) is undefined when \(z = 0\). Therefore, the value \(z = 0\) is not in the domain.