Problem 8 - A Graphical Approach to College Algebra

Question

Find the domain of each rational expression. $$\frac{9 x+12}{(2 x+3)(x-5)}$$

Step-by-Step Solution

Answer

The domain is \((-\infty, -\frac{3}{2}) \cup (-\frac{3}{2}, 5) \cup (5, \infty)\).

Step 1: Understand the Rational Expression

A rational expression is a fraction of two polynomials. In this case, the expression is \( \frac{9x+12}{(2x+3)(x-5)} \). The domain of this expression consists of all real numbers except those that make the denominator zero.

Step 2: Identify the Denominator

The denominator of the given expression is \((2x+3)(x-5)\). We must find the values of \(x\) that make this expression equal to zero.

Step 3: Set Denominator Equal to Zero

To find the values that are not in the domain, solve \((2x+3)(x-5) = 0\). This means either \(2x+3 = 0\) or \(x-5 = 0\).

Step 4: Solve Each Equation Separately

Solve \(2x+3 = 0\):Subtract 3 from both sides to get \(2x = -3\). Then divide both sides by 2 to solve for \(x\):\(x = -\frac{3}{2}\).Solve \(x-5 = 0\):Add 5 to both sides to get \(x = 5\).

Step 5: Combine Exclusion Values

The values \(-\frac{3}{2}\) and \(5\) are not in the domain because they make the denominator equal to zero. Therefore, the domain of the expression excludes these two values.

Step 6: Write the Domain in Interval Notation

The domain of the expression is all real numbers except \(-\frac{3}{2}\) and \(5\). In interval notation, this is:\(( -\infty, -\frac{3}{2} ) \cup ( -\frac{3}{2}, 5 ) \cup (5, \infty )\).

Key Concepts

Domain of a Function

In mathematics, the domain of a function refers to all the possible input values (usually represented by 'x') that the function can accept without causing any mathematical errors. When dealing with rational expressions, identifying the domain involves ensuring that the denominator of the expression doesn't equal zero. This is because division by zero is undefined in mathematics.
For the expression \(\frac{9x+12}{(2x+3)(x-5)}\), we looked at the denominator, which is \((2x+3)(x-5)\), to find values of \(x\) that make it zero.

• If \(2x+3=0\), then \(x=-\frac{3}{2}\).
• If \(x-5=0\), then \(x=5\).

These values are excluded from the domain because they cause the denominator to become zero. Understanding the domain of a function ensures that we are only choosing input values that result in a valid and meaningful output.

Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simple terms, it's an expression like \(ax^n + bx^{n-1} + ... + c\), where \(a, b,\) and \(c\) are coefficients, and \(n\) is a non-negative integer.
The expression \(9x + 12\) in our problem is a polynomial. It is of degree one, often called a linear polynomial. The denominator \((2x+3)(x-5)\) is also formed using simple polynomials multiplied together.
Polynomials are fundamental in algebra, as they form the building blocks of more complex mathematical concepts. They have straightforward rules for operations such as addition, subtraction, and multiplication, which makes them easy to manipulate while simplifying rational expressions.

Interval Notation

Interval notation is a concise way of describing a set of numbers along a continuous range on the number line. It is used to specify the domain of a function succinctly and is especially useful when excluding certain values, as seen in rational expressions.
In interval notation:

• Round brackets \(( )\) signify that an endpoint is not included (known as an "open" interval).
• Square brackets \([ ]\) indicate that an endpoint is included (a "closed" interval).

For the given rational expression, the domain excludes \(-\frac{3}{2}\) and \(5\), written in interval notation as \(( -\infty, -\frac{3}{2} ) \cup ( -\frac{3}{2}, 5 ) \cup (5, \infty )\).
This tells us that all real numbers are included in the domain, except \(-\frac{3}{2}\) and \(5\). Understanding and using interval notation allows for a clear and efficient way of expressing complex domain restrictions.