Question

Find the domain of each function.\(f(x)=\frac{x-1}{|3 x-1|-4}\)

Short answer

Domain: \((-\infty, -1) \cup (-1, \frac{5}{3}) \cup (\frac{5}{3}, \infty)\)

Step-by-step solution

Identify the possible restrictions on the domain

The denominator of the given function is \(|3x - 1| - 4\). Since division by zero is undefined, the expressions where the denominator equals zero must be excluded from the domain.

Set the denominator to zero

To find where the denominator is zero, solve the equation \(|3x - 1| - 4 = 0\).

Solve the absolute value equation

Rewrite the equation as \(|3x - 1| = 4\). This results in two separate equations: \(3x - 1 = 4\) and \(3x - 1 = -4\).

Solve for \(x\)

For the equation \(3x - 1 = 4\), solve for \(x\): \(3x = 5 \rightarrow x = \frac{5}{3}\). For the equation \(3x - 1 = -4\), solve for \(x\): \(3x = -3 \rightarrow x = -1\).

Exclude these values from the domain

Since the denominator becomes zero at \(x = \frac{5}{3}\) and \(x = -1\), these values must be excluded from the domain.

Write the domain of the function

The domain of the function \(f(x) = \frac{x-1}{|3 x-1|-4}\) includes all real numbers except \(x = \frac{5}{3}\) and \(x = -1\). In interval notation, the domain can be written as \((-\infty, -1) \cup (-1, \frac{5}{3}) \cup (\frac{5}{3}, \infty)\).

Key concepts

Absolute Value Equations

An absolute value equation involves an absolute value expression. The absolute value of a number represents the distance from zero on the number line, regardless of direction. For example, \(|-5| = 5\) and \(|5| = 5|\). The absolute value equation you encounter in this exercise is \[|3x - 1| - 4 = 0.\]. When solving absolute value equations, you typically rewrite the absolute value portion to solve for two separate cases: positive and negative. Here, it means solving both \(|3x - 1| = 4\) and \(|3x - 1| = -4\). This splits into two simple linear equations: \[3x - 1 = 4 \] and \[3x - 1 = -4.\]. By doing so, you find two solutions: \[x = \frac{5}{3}\] and \[x = -1.\]. These values show points where the denominator would become zero, impacting the domain.

Undefined Expressions

Undefined expressions in math occur when certain operations can't produce a valid result. One common situation leading to undefined expressions is division by zero. For a function to be defined at every point in its domain, you must ensure its denominator never equals zero.
In this exercise, you must check where the denominator \[|3x - 1| - 4\] equals zero. If \[|3x - 1| - 4\] were zero, it would make the function undefined. So, you solve for points where the expression seems undefined and exclude those from the domain to maintain a well-defined function. By setting the denominator to zero and solving as discussed earlier, you find these critical points are \[x = \frac{5}{3}\] and \[x = -1.\]. Excluding these points ensures a defined function across its domain.

Interval Notation

Interval notation is a concise way to describe subsets of real numbers. When describing the domain of a function, you often rely on interval notation to exclude certain points where the function is undefined.
In our exercise, after finding that the function \(|3x - 1| - 4\) is undefined at \[x = \frac{5}{3}\] and \[x = -1\], you express the valid domain in interval notation. You do this by breaking the real number line into segments and using union symbols to combine them. The domain becomes \((-\infty, -1) \cup (-1, \frac{5}{3}) \cup (\frac{5}{3}, \infty )\), which means every real number except \[x = \frac{5}{3}\] and \[x = -1\] is included. This clear notation helps avoid undefined points and confidently describes where the function is usable.

Division by Zero

Division by zero is a major issue in mathematics, creating undefined expressions. Any number divided by zero doesn't result in a valid number but rather an undefined scenario. This concept is crucial for understanding function behavior.
When analyzing the function \[f(x) = \frac{x-1}{|3x - 1| - 4},\] you must ensure the denominator \[|3x - 1| - 4\] never equals zero. If it did, you encounter division by zero, meaning the function is undefined at those points. Solving \[|3x - 1| - 4 = 0\] reveals this occurs at \[x = \frac{5}{3}\] and \[x = -1.\] Excluding these values from the domain avoids undefined divisions. Always check for zero in denominators within functions to ensure their proper definition and usage.