Problem 111 - College Algebra

Question

Find the domain of each logarithmic function. $$ f(x)=\log \left(\frac{x+1}{x-5}\right) $$

Step-by-Step Solution

Answer

The domain of the function \( f(x)=\log \left(\frac{x+1}{x-5}\right) \) is \( (-\infty, -1) \cup (5, \infty) \).

Step 1: Identify the input to the log function

Identify the argument of the logarithmic function. In this case, it is \( \frac{x+1}{x-5} \). For a logarithm to be real, the argument of the logarithm should be greater than zero.

Step 2: Set up the inequality

For the fraction to be more than zero, either both the numerator and denominator should be more than zero, or both should be less than zero. Set up the inequality \( \frac{x + 1}{x - 5} > 0 \), and this will give two conditions : (1) \( x + 1 > 0 \) and \( x - 5 > 0 \) or (2) \( x + 1 < 0 \) and \( x - 5 < 0 \).

Step 3: Solve the inequality

Solving the inequality gives two possible ranges of x: (1) \( x > - 1 \) and \( x > 5 \) that combines to \( x > 5 \) or (2) \( x < -1 \) and \( x < 5 \) combining to \( x < -1 \). Also, remember that denominator of the fraction can't be equal to zero so, exclude \( x = 5 \) from the domain.

Step 4: State the domain

Combine these intervals to give the domain of the function, which is \( x < -1 \) or \( x > 5 \). Written in interval notation, the domain is \( (-\infty, -1) \cup (5, \infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to all possible input values ("x" values) the function can accept without causing any mathematical errors. When dealing with logarithmic functions, determining the domain can be tricky but it's crucial for ensuring the function remains valid.
In the case of a function like \( f(x) = \log \left( \frac{x+1}{x-5} \right) \), the domain is determined by analyzing the expression inside the log function. A logarithm is real only when its argument is greater than zero.
Thus, for our specific function, it means that \( \frac{x+1}{x-5} > 0 \). By ensuring that this fraction respects the conditions where a fraction is greater than zero, we identify the domain limits. Don't forget, division by zero is not allowed, so we need to make sure \( x eq 5 \).

Inequalities

Inequalities are used to determine the conditions under which our logarithmic expression is valid. An inequality can tell us where the expression inside a function like \( f(x) = \log \left( \frac{x+1}{x-5} \right) \) is positive, negative, or zero.
In our problem, we set up the inequality \( \frac{x + 1}{x - 5} > 0 \). This inequality helps us understand that the function's argument must be positive.

• If both numerator \(x+1\) and denominator \(x-5\) are positive, the overall fraction is positive.
• If both are negative, the fraction is still positive.

Solving leads us to identify two separate intervals for \( x \): either "x" is greater than 5, or "x" is less than -1. Remember, at no point can \( x-5 = 0 \) since division by zero is undefined.

Interval Notation

Once we determine the viable intervals for \( x \), we need a concise way to express them, and that's where interval notation comes in. Interval notation is a simplified way to describe continuous sequences of numbers without listing every single one.
From our inequality analysis, we found the domain of \( f(x) = \log \left( \frac{x+1}{x-5} \right) \) to be two ranges: \( x < -1 \) and \( x > 5 \).
In interval notation, these are expressed as \( (-\infty, -1) \) and \( (5, \infty) \) respectively.
We use a union sign (\( \cup \)) to show that either of these intervals satisfies the domain, giving us a complete notation: \( (-\infty, -1) \cup (5, \infty) \).