Problem 45 - Contemporary Precalculus

Question

Find the domain of the given function (that is, the largest set of real numbers for which the rule produces well-defined real numbers). $$h(x)=\log (-x)$$

Step-by-Step Solution

Answer

Answer: The domain of the function $$h(x)=\log(-x)$$ is $$(-\infty,0)$$.

Step 1: Logarithm properties

Recall that the logarithm function is only well-defined for positive arguments. This means we need to find the values of x such that $$-x$$ is positive.

Step 2: Solve the inequality

To find the domain of the function, we must solve the inequality $$-x > 0$$ or equivalently, we can multiply both sides by $$-1$$, remembering that we need to reverse the inequality sign: $$x < 0$$.

Step 3: Write the domain

The solution to the inequality gives us the domain for the function $$h(x)=\log(-x)$$. The domain is $$x<0$$, or in interval notation, we can write $$(-\infty,0)$$.

Key Concepts

Logarithmic Functions

Logarithmic functions are fascinating mathematical tools that express how many times a certain number, called the base, must be multiplied by itself to reach another number. The general form of a logarithmic function is \( \log_b(x) \), where \( b \) is the base, and \( x \) is the argument of the logarithm. In practice, we most often use common logarithms (base 10) or natural logarithms (base \( e \)).
Logarithmic functions have specific properties that determine where they are defined. One crucial property is that the argument of a logarithm must always be positive. This means we cannot take the logarithm of zero or a negative number, as it is not defined in the real number system. This property is key when determining the domain of logarithmic functions in mathematical problems.
In the problem, we have the function \( h(x) = \log(-x) \). To ensure the argument \( -x \) is positive, \( x \) must be negative. This restricts \( x \) so that it only includes numbers less than zero. Understanding this property helps in identifying the domain of any logarithmic function.

Inequalities

Inequalities are mathematical expressions indicating that two values are not necessarily equal and instead describe a range of possible values. They play an essential role when dealing with domains of functions, especially those involving logarithmic functions.
When solving inequalities, we often isolate the variable on one side to determine its valid range. In the context of logarithmic functions, we use inequalities to find conditions where the argument remains positive. Working with inequalities involves flipping the inequality sign when multiplying or dividing by a negative number.
In the function \( h(x) = \log(-x) \), we need \( -x > 0 \) for the expression to be defined. By solving this inequality, we find that \( x < 0 \). Thus, inequalities guide us in defining appropriate domains for functions, helping ensure they remain valid across their entire range.

Real Numbers

Real numbers form the backbone of many mathematical concepts, including the study of functions. They include all the numbers on the number line, comprising both rational numbers (like fractions) and irrational numbers (such as \( \pi \) or \( \sqrt{2} \)). Real numbers exclude imaginary numbers, which are rooted in the concept of \( \sqrt{-1} \), also known as \( i \).
When determining the domain of a function, we are focused on finding the largest set of real numbers where the mathematical rule can be applied to produce a well-defined function. Real numbers help limit and define function domains by excluding any values that would make the function undefined, such as negatives in logarithmic arguments or zero in denominators.
In this specific case, the domain of \( h(x) = \log(-x) \) is the subset of real numbers \( x < 0 \), which ensures that all values used in the function produce valid, defined results. Understanding real numbers and their properties is fundamental to solving equations and defining functions in mathematics.