Problem 41 - Precalculus with Calculus Previews

Question

In Problems \(39-44\), find the domain of the given function \(f\). $$ f(x)=\ln \left(9-x^{2}\right) $$

Step-by-Step Solution

Answer

The domain of the function is \((-3, 3)\).

Step 1: Understanding Domain of a Logarithmic Function

The domain of a logarithmic function, such as \( f(x) = \ln(9-x^2) \), requires the argument inside the logarithm (i.e., \(9-x^2\)) to be greater than 0 because the logarithm of zero or a negative number is undefined.

Step 2: Setting Up the Inequality

For the function \( f(x) = \ln(9-x^2) \), we need to solve the inequality \(9-x^2 > 0\). This will determine the values of \(x\) for which the function is defined.

Step 3: Solving the Inequality

Rearrange the inequality \(9 - x^2 > 0\) to \(9 > x^2\). This further simplifies to \(x^2 < 9\).

Step 4: Finding the Solution for x

Solve \(x^2 < 9\) by taking the square root of both sides. This gives \(-3 < x < 3\). Hence, the domain of \(f(x) = \ln(9-x^2)\) is the interval \((-3, 3)\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions involving symbols like "<", ">", "≤", or "≥". They are used to define ranges of possible values that satisfy a given condition. In solving inequalities, the goal is to find the set of values, often in terms of a variable, that make the inequality true. Inequalities play a crucial role when determining the domain of functions, particularly when restrictions appear due to the function's form.

When solving an inequality, we often apply algebraic manipulations much like we do in an equation. However, it's pivotal to remember a distinct rule: multiplying or dividing both sides of an inequality by a negative number flips the inequality sign. This rule is specific to inequalities and is an important detail to keep in mind.

In the context of logarithmic functions, inequalities detail the range of acceptable values for the argument inside the logarithm. For example, in the function \( f(x) = \ln(9-x^2) \), the inequality \( 9-x^2 > 0 \) is crucial – it tells us under what conditions the logarithm is defined, thus steering us towards finding the function's domain.

Determining the Function Domain

The function domain refers to all the possible values of the input variable (commonly \( x \)) for which the function is defined and yields a real number output. Finding the domain of a function is fundamental as it sets the limits within which the function operates and produces meaningful results.

• For basic algebraic and polynomial functions, the domain often includes all real numbers unless otherwise specified by denominators that could become zero or radicals requiring non-negative arguments.
• For logarithmic functions, a stricter condition applies: the argument inside the logarithmic function must be greater than zero, as logarithms for zero or negative numbers are undefined in the real number system.

To determine the domain of the logarithmic function \( f(x) = \ln(9-x^2) \), we set the argument \( 9-x^2 \) greater than zero, forming the inequality \( 9 - x^2 > 0 \). Solving this inequality gives us the set of permissible \( x \) values where the function remains valid and returns a real number, leading to the domain: \(-3 < x < 3\).

Exploring Logarithmic Functions

Logarithmic functions are a vital part of mathematics, particularly in fields dealing with exponential growth or decay, such as biology, economics, and physics. A logarithmic function typically takes the form \( f(x) = \ln(a) \), where \( a \) is the argument. The natural logarithm, denoted as \( \ln \), is the inverse of the exponential function, \( e^x \).

Some key characteristics of logarithmic functions are:

• They are defined only for positive arguments, which means the domain is restricted to values greater than zero.
• At the point where the argument equals 1, the logarithm is zero because \( \ln(1) = 0 \).
• As the argument approaches infinity, the logarithm approaches infinity as well.
• Importantly, as the argument approaches zero from the positive side, the logarithm tends toward negative infinity.

Understanding these properties is crucial when working with logarithmic functions, especially when identifying the domain. For \( f(x) = \ln(9-x^2) \), we only consider values of \( x \) that ensure the argument \( 9-x^2 \) remains positive to avoid undefined or non-real results.