Question

Determine if each value of \(x\) is in the domain of the expression.\(\frac{2 x+3}{x-4} \quad\) (a) \(x=-\frac{3}{2} \quad\) (b) \(x=4\)

Short answer

The value \(x = -\frac{3}{2}\) is in the domain of the function, while \(x = 4\) is not.

Step-by-step solution

Substitute the given values of \(x\)

First, for \(x = -\frac{3}{2}\), substitute this value to the function and see if it produces a defined value: \[\frac{2(-\frac{3}{2}) + 3}{-\frac{3}{2} - 4}\] This simplifies to \(\frac{-3 + 3}{-\frac{11}{2}} = 0\). Since division by zero does not occur, it means \(x = -\frac{3}{2}\) is within the domain of the function.

Check for the value \(x = 4\)

For \(x = 4\), substitute this value to the function and see if it produces a defined value: \[\frac{2(4) + 3}{4 - 4}\] This simplifies to \(\frac{11}{0}\) and it is undefined. Thus, \(x = 4\) is not within the domain of the function.

Key concepts

Understanding Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. It involves substituting a specific value into the variable of a function. In the context of our exercise, the function given is a rational expression: \( \frac{2x+3}{x-4} \). Here, we're tasked with determining if certain values are within the domain of this expression by substituting these values into the rational function.
For example, for \( x = -\frac{3}{2} \), we substitute \( -\frac{3}{2} \) in place of \( x \). Evaluating this gives:\[ \frac{2(-\frac{3}{2}) + 3}{-\frac{3}{2} - 4} = \frac{-3 + 3}{-\frac{11}{2}} = 0 \] This evaluation shows that at \( x = -\frac{3}{2} \), the function gives an output of 0, which means this value is within the domain, as it results in a defined value and no division by zero occurs.
After every evaluation, it is critical to examine if the results are sensible and valid by checking for any undefined terms, like division by zero.

Avoiding Division by Zero

Division by zero is a common issue in mathematics, leading to undefined expressions. It occurs when a function denominator equals zero, rendering the expression invalid. In rational functions, it's crucial to identify values of \( x \) that cause the denominator to be zero.
In the function \( \frac{2x+3}{x-4} \), the denominator is \( x - 4 \). To avoid division by zero, \( x \) must not equal 4. This would cause the denominator to be zero, resulting in the undefined expression: \( \frac{11}{0} \).
Thus, when evaluating functions, always check the denominator. If substituting a value results in zero in the denominator, this signals a restriction in the domain. Avoid these values to ensure the function remains valid, as seen in our exercise where \( x = 4 \) leads to division by zero.

The Nature of Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Just like regular fractions, the key consideration for rational expressions is to ensure the denominator is never zero.
This can be thought of as the core rule while handling rational expressions. Below are some points to note:

• The domain of a rational expression is all real numbers except where the denominator is zero.
• Always simplify the expression if possible for easier evaluation.
• Factor the denominator if needed to identify problematic \( x \) values more easily.

For the expression \( \frac{2x+3}{x-4} \), identifying where \( x - 4 = 0 \) (hence \( x = 4 \)) points out values that must be excluded from the domain. By understanding these properties and steps, students can handle rational expressions confidently, ensuring they do not fall into the trap of division by zero and can determine the domain accurately.