Problem 37 - Elementary and Intermediate Algebra

Question

Find the domain of \(f\) $$ f(x)=\frac{x-5}{2 x+1} $$

Step-by-Step Solution

Answer

The domain of \( f(x) = \frac{x-5}{2x+1} \) is all real numbers except \( x = -\frac{1}{2} \).

Step 1: Understand the Function

The function given is \[ f(x) = \frac{x - 5}{2x + 1} \]. This is a rational function, which is a fraction where the numerator and the denominator are both polynomials.

Step 2: Identify Restrictions for the Domain

For rational functions, the domain is all real numbers except where the denominator is zero. To find where the denominator is zero, set the denominator equal to zero and solve for \(x\).

Step 3: Solve for the Prohibited Values

Set the denominator equal to zero and solve for \(x\): \[ 2x + 1 = 0 \]Subtract 1 from both sides: \[ 2x = -1 \]Divide both sides by 2: \[ x = -\frac{1}{2} \]

Step 4: State the Domain

Since \( x = -\frac{1}{2} \) makes the denominator zero, it must be excluded from the domain. Therefore, the domain of \( f(x) \) is all real numbers except \( x = -\frac{1}{2} \).

Step 5: Express the Domain in Interval Notation

The domain in interval notation is: \( (-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty) \)

Key Concepts

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Rational functions are fractions where both the top and the bottom parts (called the numerator and denominator) are polynomials. In simpler terms, they're like regular fractions, but their components involve variables like \(x\).

• The numerator can be any polynomial.
• The denominator must also be a polynomial but cannot be zero.

These functions often appear in algebra and calculus, and they have specific rules for determining their domains.

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Denominator restrictions are crucial when working with rational functions. The denominator is the bottom part of the fraction in these functions. Since division by zero is undefined in mathematics, you must determine which values of \(x\) make the denominator equal to zero and exclude them from the domain.

To find the restricted values:

• Set the denominator equal to zero.
• Solve the resulting equation for \(x\).

For the function \(f(x) = \frac{x-5}{2x+1}\), the denominator is \(2x + 1\).

To find when it is zero, solve \(2x + 1 = 0\). Subtract 1 from both sides to get \(2x = -1\). Divide both sides by 2 to get \(x = -\frac{1}{2}\). So, \(x = -\frac{1}{2}\) is the restricted value and must be excluded from the domain.

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Interval notation is a way to describe intervals on the real number line. This is especially useful when representing domains after restrictions have been applied.

In interval notation:

• Parentheses \(()\) are used to indicate that an endpoint is not included.
• Brackets \([]\) are used to indicate that an endpoint is included.
• The union symbol \(\cup\) is used to combine separate intervals.

For the function \(f(x) = \frac{x-5}{2x+1}\), after excluding \(x = -\frac{1}{2}\), the domain is all real numbers except \(-\frac{1}{2}\).

In interval notation, this is written as: \((-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)\). This means the function is defined for all \(x\) in the interval from negative infinity to negative one-half (not including \(-\frac{1}{2}\)), and from negative one-half to positive infinity.