Question

Find the domain of each function. \(p(x)=\frac{x}{|2 x+3|-1}\)

Short answer

The domain is \( \{ x \in \mathbb{R} \,|\, x e -1 \,\text{and}\, x e -2 \} \).

Step-by-step solution

- Identify the function form

Observe that the function is given as \( p(x) = \frac{x}{|2x+3| - 1} \). Recognize that the function consists of a numerator, \(x\), and a denominator, \(|2x + 3| - 1\).

- Determine when the denominator is defined

In order for the function \( p(x) \) to be defined, the denominator, \(|2x + 3| - 1\), must not be zero. Therefore, find values of \(x\) for which \(|2x + 3| - 1 = 0\).

- Solve the equation for the denominator

Set the denominator equal to zero and solve for \(x\):\[ |2x + 3| - 1 = 0 \]Adding 1 to both sides:\[ |2x + 3| = 1 \]This gives us two equations to solve:\[ 2x + 3 = 1 \] and \[ 2x + 3 = -1 \]

- Solve each equation separately

Solve the first equation:\[ 2x + 3 = 1 \]Subtract 3 from both sides:\[ 2x = -2 \]Divide by 2:\[ x = -1 \]Solve the second equation:\[ 2x + 3 = -1 \]Subtract 3 from both sides:\[ 2x = -4 \]Divide by 2:\[ x = -2 \]

- Combine the results

The values \( x = -1 \) and \( x = -2 \) make the denominator zero, causing the function to be undefined. Therefore, the domain of the function is all real numbers except \( x = -1 \) and \( x = -2 \).

- Write the domain in set notation

Express the domain of the function in set notation:\( \text{Domain}: \{ x \in \mathbb{R} \,|\, x e -1 \,\text{and}\, x e -2 \} \)

Key concepts

Denominator

In a fraction, the denominator is the bottom part of the fraction. It tells us how many parts the whole is divided into. For example, in the fraction \( \frac{3}{5} \), the denominator is \(5\). In this exercise, the function is \( p(x) = \frac{x}{|2x+3|-1} \). The denominator is \( |2x+3| - 1 \). The denominator cannot be zero because division by zero is undefined.

Absolute Value

Absolute value measures the distance a number is from zero on a number line, without considering direction. It's always a non-negative value. For example, \( | -4 | = 4 \) and \( | 4 | = 4 \), because both -4 and 4 are 4 units away from zero. In the function \( p(x) = \frac{x}{|2x+3|-1} \), \( |2x+3| \) means that whatever inside the absolute value bars, we take its non-negative distance from zero.

Set Notation

Set notation is a way to describe a set of numbers. When we talk about the domain of a function, we can use set notation to clearly list all possible values of \( x \). For example, if \( x \) can be any number except -1 and -2, we write this as: \( \{ x \in \mathbb{R} \,|\, x eq -1 \text{ and } x eq -2 \} \). This means 'the set of all real numbers \( x \) such that \( x \) is not equal to -1 and -2'.

Undefined Values

Values that make a function undefined are values for which the function cannot produce a result. For the function \( p(x) = \frac{x}{|2x+3|-1} \), if the denominator \( |2x + 3| - 1 \) equals zero, the function is undefined. Solving equations \( 2x + 3 = 1 \) and \( 2x + 3 = -1 \), we find \( x = -1 \) and \( x = -2 \) respectively. These values make the denominator zero, causing the function to be undefined. Hence, the domain excludes these values: the function is defined for all \( x \) except \( x = -1 \) and \( x = -2 \).