Problem 25 - Elementary Algebra

Question

What is the domain of \(y=\frac{3 x-7}{x+3}\) ?

Step-by-Step Solution

Answer

Answer: The domain of the function is \(\{x \in \mathbb{R} : x \ne -3\}\), which means the function is valid for all real numbers except x = -3.

Step 1: Identify the potential issue of the function

In this exercise, the main concern when determining the domain is the denominator, as we cannot have division by zero. So, the denominator cannot be equal to zero.

Step 2: Set the denominator equal to zero and solve

To find the value of x where the denominator becomes zero, set the denominator equal to zero: \(x + 3 = 0\) Now, subtract 3 from both sides: \(x=-3\)

Step 3: Write the domain

Since the function becomes undefined at x = -3, the domain will consist of all real number x-values except -3. We can write the domain as: Domain = \(\{x \in \mathbb{R} : x \ne -3\}\)

Key Concepts

Rational Functions

A rational function is a type of function that is represented by the ratio of two polynomials. In other words, it has the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) \) is not the zero polynomial. The domain of a rational function includes all real numbers except those that make the denominator \( q(x) \) equal to zero, since division by zero is undefined.

The function provided in the exercise \( y=\frac{3x-7}{x+3} \) is a rational function. Here, \( p(x) = 3x - 7 \) and \( q(x) = x + 3 \) are polynomials. To find the domain of this function, we focus on the denominator \( q(x) \) and identify values of \( x \) that would make it zero, since those values cannot be included in the domain of the function. Understanding the nature of rational functions is key to grasping concepts in algebra and calculus, where these functions often appear.

Division by Zero

Division by zero is a concept in mathematics that is crucial to understand as it is not allowed and is considered undefined. In any mathematical operation, if the divisor (the number you're dividing by) is zero, the operation does not have a real number result. A common misconception is that dividing by zero might yield infinity, but in reality, the result is undefined because it breaks the fundamental properties of arithmetic.

For example, with the rational function \( y=\frac{3x-7}{x+3} \), we cannot allow \( x = -3 \) because it would result in \( x+3 = 0 \) in the denominator, which leads to division by zero. To avoid this, we exclude \( x = -3 \) from the function's domain. Understanding why division by zero is not possible helps students avoid making mistakes when solving equations and finding the domains of functions.

Solving Equations

Solving equations is a fundamental skill in mathematics. An equation represents a statement that two expressions are equal, and solving an equation involves finding the value(s) of the variable(s) that make that statement true. The process often involves isolating the variable using algebraic operations, ensuring that the operations performed on one side of the equation are also done to the other side to maintain equality.

In the context of rational functions, as seen in the exercise \( y=\frac{3x-7}{x+3} \), solving the equation \( x+3=0 \) was necessary to identify the value that cannot be in the domain. To solve for \( x \) in this case, we subtract 3 from both sides of the equation, yielding the solution \( x=-3 \). Equations can vary in complexity, and the steps taken to solve them will depend on their form. Whether dealing with linear, quadratic, or more complex polynomials, the goal remains the same: to find the value(s) of the variable(s) that satisfy the equation.