Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$f(x)=\log (-x)$$

Step-by-Step Solution

Answer

The domain of \( f(x) = \log(-x) \) is \((-\infty, 0)\).

Step 1: Understand the Function

The function given is \( f(x) = \log(-x) \). To find the domain of this function, we need to ensure the expression inside the logarithm is positive.

Step 2: Define the Condition for Logarithms

Recall that for \( \log(b) \) to be defined, \( b \) must be greater than zero. Therefore, \( -x \) must be greater than zero for \( f(x) = \log(-x) \) to be defined.

Step 3: Solve the Inequality

Based on the inequality from Step 2, solve \( -x > 0 \). This simplifies to \( x < 0 \).

Step 4: Express the Domain

The inequality \( x < 0 \) tells us that the domain of \( f(x) = \log(-x) \) consists of all real numbers less than zero. In interval notation, the domain is \( (-\infty, 0) \).

Key Concepts

Understanding the Domain of Logarithmic Functions

When we talk about the domain of a function, we're discussing the set of all possible input values (usually represented as 'x') that the function can accept without causing any mathematical issues. In logarithmic functions, like the one given as \( f(x) = \log(-x) \), there are specific rules for what 'x' can be, because not every real number will work. For a logarithmic function \( \log(b) \), the expression inside the log, 'b,' must be a positive number. This is because the logarithm is only defined for positive values of 'b.' Therefore, to determine the domain of our function, we make sure the expression inside the logarithm remains positive. For our function, this means determining the values of 'x' for which \( -x > 0 \). By solving this inequality, it becomes clear why 'x' must be less than zero, meaning that the domain is all real numbers less than zero. We express this using interval notation as \( (-\infty, 0) \).

Solving Inequalities in Logarithms

To find the domain of a logarithmic function, you need to tackle inequalities. Inequalities help us figure out the range of acceptable 'x' values. In the context of \( f(x) = \log(-x) \), the inequality to solve is \( -x > 0 \). This inequality arises because the argument (the expression inside the logarithm) must be positive for the log function to be defined.Here's how we solve it:- Start with \( -x > 0 \).- Multiply both sides by -1, but remember to flip the inequality sign. This gives us \( x < 0 \).This tells us that 'x' must be any value less than zero. Understanding how inequalities switch directions when multiplied or divided by negative numbers is crucial in solving these kinds of problems.

Expressing Domain in Interval Notation

Once you've determined the values 'x' can take, it’s time to express the domain in interval notation. Interval notation is a concise way to represent ranges of numbers using brackets and parentheses. It tells us exactly which numbers are included in a set.For our function \( f(x) = \log(-x) \), the solution to the inequality \( x < 0 \) reveals the domain. In interval notation, this is written as \( (-\infty, 0) \). Here's what this means:- The parenthesis '(' and ')' are used because neither -∞ nor 0 are included in the domain.- -∞ represents the idea that there is no lower limit to 'x'; it can be any negative number.- 0 is not included since the expression \(-x\) must be positive.Using interval notation makes it easy to quickly communicate the range of acceptable values for a function’s domain.