Problem 1 - Algebra and Trigonometry

Question

Find the domain of each rational function. \(f(x)=\frac{5 x}{x-4}\)

Step-by-Step Solution

Answer

The domain of \(f(x) = \frac{5x}{x-4}\) is all real numbers except \(x = 4\).

Step 1: Identify the denominator

The denominator of the rational function is \(x - 4\).

Step 2: Equate the denominator to zero

The next step is to find the value of 'x' where the denominator would become zero. This can be done by setting the denominator equal to zero and solve for 'x'. So, \(x-4=0\). Solving this gives \(x = 4\).

Step 3: Set the domain

Since we have found that \(x = 4\) makes the denominator zero, and division by zero is undefined, the domain of the function, which is the set of all 'x' for which the function is defined, is all real numbers except x = 4.

Key Concepts

Rational Functions

A rational function is a type of function that is the ratio of two polynomials. It is expressed in the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials. The most significant feature of rational functions is that they may have values of \( x \) for which the function is not defined, specifically where the denominator, \( q(x) \) equals zero.

Rational functions can take on a wide range of behaviors, such as asymptotic behavior, where the function's graph tends toward a line without touching it, and discontinuities, where the function is not continuous. These functions model many real-world situations, like the ratio of two changing quantities. Understanding how to work with rational functions includes knowing how to find their domain, which is essentially identifying the values for which the function is mathematically valid.

Identifying Domain

The domain of a function is the complete set of possible values of the independent variable, which in most cases is \( x \). For rational functions, the domain consists of all the real numbers except those that make the denominator equal to zero. To identify the domain,
1. Observe the denominator of the rational function.
2. Set the denominator equation equal to zero and solve for \( x \).
3. Exclude the resulting \( x \) values from the real numbers.

These steps are crucial because they prevent the undefined scenario of division by zero. Having a clear understanding of how to find the domain of a function is fundamental in mathematics as it informs you of the acceptable inputs that can be inserted into the function, ensuring the output remains meaningful and accurate.

Division by Zero

In mathematics, division by zero is a significant concept because it is an operation that is undefined. This means there is no number that you can multiply by zero to get a nonzero number, which makes any fraction with zero in the denominator non-computable. As such, any value that would cause the denominator of a rational function to become zero must be excluded from the domain.

Why is division by zero undefined? Let's say we could divide by zero and get a number \( a \), then \( 0 \times a \) should be equal to the numerator of the original fraction. However, \( 0 \times a = 0 \) for any number \( a \) thus making it impossible to satisfy the condition. The concept of division by zero is not just an abstract mathematical rule but a fundamental principle that maintains the consistency of arithmetic operations and their properties. Hence, it is one of the first checks done when determining the domain of a rational function.