Problem 43 - 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). $$f(x)=\ln (x+1)$$

Step-by-Step Solution

Answer

Answer: The domain of the function \(f(x) = \ln(x+1)\) is \((-1, \infty)\), representing all real numbers x strictly greater than -1.

Step 1: Understand the constraints of the natural logarithm

For the natural logarithm function \(\ln(x)\) to be defined, the argument x must be strictly greater than 0 (i.e., \(x > 0\)). In our case, the argument is \((x + 1)\), so we will need to find the range of x values such that \((x+1)>0\).

Step 2: Set up the inequality

Set up an inequality with the argument: $$(x+1) > 0$$

Step 3: Solve the inequality

Solve the inequality for x: $$x > -1$$ Now, the inequalities tell us that the domain of the function \(f(x) = \ln(x+1)\) is all real numbers x such that \(x > -1\).

Step 4: Write down the domain in interval notation

Using interval notation, the domain of the function can be written as: $$(-1, \infty)$$ This represents all real numbers x strictly greater than -1.

Key Concepts

Interval Notation

Interval notation is a mathematical way of describing a set of numbers along a number line. It provides a concise and clear method to specify the range of values that an expression can take. In our exercise, after determining the inequality \(x > -1\), we use interval notation to express the domain of the function \(f(x) = \ln(x+1)\).

In interval notation, brackets and parentheses are used to show whether endpoints are included or excluded from the set:

• An open interval \((a, b)\) indicates that the endpoints \(a\) and \(b\) are not included in the set.
• A closed interval \([a, b]\) includes both endpoints.
• A half-open (or half-closed) interval like \((a, b]\) or \([a, b)\) includes one endpoint but not the other.

For the domain \(x > -1\), the interval notation is \((-1, \infty)\). This represents all real numbers greater than \(-1\) up to infinity, excluding \(-1\) itself.

Natural Logarithm

The natural logarithm is a fundamental mathematical function with a base of \(e\), where \(e \approx 2.71828\). It is denoted by \(\ln(x)\) and is the inverse operation of exponentiation involving \(e\). This means that if \(y = \ln(x)\), then \(x = e^y\).

Understanding the natural logarithm is crucial for solving problems where exponential growth or decay is present. For example, in the context of the function \(f(x) = \ln(x+1)\), the natural logarithm helps us understand the behavior of the function as \(x\) changes.

Key properties of the natural logarithm include:

• The domain is \((0, \infty)\), meaning it can only operate on positive numbers.
• It passes through the point \((1, 0)\), since \(\ln(1) = 0\).
• It is undefined for \(x \leq 0\), reinforcing the importance of recognizing constraints in algebraic functions.

In our exercise, it is important to understand why \(x+1 > 0\) must hold, as it ensures the input is suitable for the natural logarithm.

Inequality

Inequalities are mathematical expressions that show the relationship between two values when they are not equal to each other. Using inequalities, we can define ranges for which certain conditions hold true. In our exercise, we used the inequality \((x + 1) > 0\) to determine the permissible values of \(x\) for the function \(f(x) = \ln(x+1)\).

There are several symbols and terminology commonly used with inequalities:

• "\(>\)" means "greater than."
• "\(<\)" means "less than."
• Adding an equals sign (\(\geq\) or \(\leq\)) includes the equality, indicating "greater than or equal to" or "less than or equal to."

Solving an inequality involves finding all values of a variable that make the inequality true. For instance, solving \((x + 1) > 0\) shows \(x > -1\). This solution gives us critical insight into which values are valid inputs for our function. By understanding these solutions, students can accurately determine domains in similar mathematical scenarios.