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Chapter 6: Problem 34

Find the domain. SEE EXAMPLE \(2 .\) (OBJECTIVE 2 ) $$\frac{12 x-7}{6 x+5}$$

Short Answer

Expert verified
The domain of the given function \(\frac{12x - 7}{6x + 5}\) is all real numbers except \(-\frac{5}{6}\), written in interval notation as \((-\infty, -\frac{5}{6}) \cup (-\frac{5}{6}, \infty)\).

Step by step solution

01

Identify the Rational Function

The provided function is a rational function given by \(\frac{12x - 7}{6x + 5}\). The domain in this case is the set of all real x-values for which this function is defined. The only limitations are the x-values that make the denominator equal to zero, because division by zero would result in an undefined function.
02

Calculate the Values of x that Make Denominator Zero

First, set the denominator of the fraction equal to zero and solve for x. This would be \(6x + 5 = 0\). Subtracting 5 from both sides gives \(6x = -5\). Then, divide both sides by 6 to solve for x, which gives \(x = -\frac{5}{6}\).
03

Identify the Domain

Knowing that x cannot be \(-\frac{5}{6}\), the domain of the function is all real numbers except \(-\frac{5}{6}\). In interval notation, this can be written as \((-\infty, -\frac{5}{6}) \cup (-\frac{5}{6}, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
A rational function is a type of function represented by the ratio of two polynomials. In simpler terms, it's a fraction where both the numerator and the denominator are polynomials. For example, if you have a polynomial in the form of \(12x - 7\), and another in the form of \(6x + 5\), putting one over the other forms our rational function: \(\frac{12x - 7}{6x + 5}\).
  • Numerator and Denominator: These are the main components of a rational function. The numerator is the top part, which in our case is \(12x - 7\), and the denominator is the bottom part, \(6x + 5\).
  • Defining Property: The critical aspect of rational functions is that they are undefined for values of \(x\) that make the denominator zero. This is because division by zero is undefined in mathematics.
To determine where a rational function is defined, one must find these values and exclude them from the domain.
Interval Notation
Interval notation is a way of writing subsets of the real number line. It provides a concise way to express solutions to inequalities and specify the domain of functions. Let's see how we use interval notation when dealing with domains of rational functions.
  • Open Intervals: The notation \((a, b)\) represents all the numbers between \(a\) and \(b\), but not including \(a\) and \(b\) themselves. The round brackets signify that the endpoints are not part of the interval.
  • Union: In cases where a function is defined over separate parts of the number line, the symbol \(\cup\) (union) is used to combine intervals. This is common for rational functions that have discontinuities.
For example, the domain \((-\infty, -\frac{5}{6}) \cup (-\frac{5}{6}, \infty)\) means that the function is defined everywhere except at \(-\frac{5}{6}\). This structure allows us to clearly indicate that \(-\frac{5}{6}\) is not part of the domain.
Undefined Expressions
Undefined expressions occur in mathematics when operations result in an impossible situation. In the context of rational functions, this happens when the denominator equals zero, as division by zero is undefined.
  • Zero in the Denominator: To determine when a rational function is undefined, solve for \(x\) in the equation where the denominator equals zero. In our function, this was done by setting \(6x + 5 = 0\).
  • Solving the Equation: By doing arithmetic operations, we find that \(x = -\frac{5}{6}\). This tells us where the function will break or be undefined.
Avoiding undefined expressions is key to correctly identifying the domain. Knowing this prevents errors and ensures we can safely execute operations within the function.

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