Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$f(x)=\ln \left(-x^{2}+16\right)$$

Step-by-Step Solution

Answer

The domain of the function is \\((-4, 4)\\).

Step 1: Understand the Domain of the Logarithmic Function

The function we are dealing with is the natural logarithm \( \ln(y) \), which is defined only for positive values of \(y\). Therefore, for \(f(x) = \ln(-x^2 + 16)\), we need \(-x^2 + 16 > 0\) to satisfy the domain condition.

Step 2: Solve the Inequality

To find the values of \(x\) for which \(-x^2 + 16 > 0\), solve the inequality: 1. Add \(x^2\) to both sides: \[-x^2 + 16 > 0 \ \ \rightarrow\ \ 16 > x^2\] 2. Rewrite it as \(x^2 < 16\) which implies: \[-4 < x < 4\]

Step 3: Express the Domain in Interval Notation

Since \(x^2 < 16\) translates to \(-4 < x < 4\), the domain of \(f(x) = \ln(-x^2 + 16)\) is written in interval notation as \((-4, 4)\). This interval comprises all the \(x\) values for which \(-x^2 + 16 > 0\).

Step 4: Verify Graphically (Optional)

We can support this result by graphing \(f(x) = \ln(-x^2 + 16)\). The graph will show that the function is only defined and produces real number outputs for \(x\) values between \(-4\) to \(4\). You can use graphing software or a graphing calculator to visualize this.

Key Concepts

Domain of a function

The domain of a function refers to the set of all possible input values (often represented by 'x') for which the function is defined. When it comes to logarithmic functions, like the natural logarithm (\( \ln(x) \)), the domain is restricted to positive numbers because the logarithm of zero or a negative number is undefined. This restriction impacts the values that 'x' can take.

For the given function \( f(x) = \ln(-x^2 + 16) \), determining the domain involves finding for which values of \( x \) the argument \( -x^2 + 16 \) is positive. Understanding the argument conditions allows us to establish where the function can actually produce a real output.

To explore the domain further:

• Start by identifying the restriction: \( -x^2 + 16 > 0 \).
• Solve this inequality to find the interval for 'x'.
• The solution gives us the valid range of inputs for the function, which is \(-4 < x < 4\).

Remember, recognizing the domain of any function is crucial to correctly using and interpreting it.

Inequalities in mathematics

Inequalities in mathematics express the relationship between two values where they are not equal, using symbols like \(<, >, \leq,\) and \(\geq\). Solving inequalities involves finding the range of values for the variables that satisfy the condition presented by the inequality.

In the context of our exercise, we look at the inequality \( -x^2 + 16 > 0 \). This inequality must be solved to determine the valid inputs for the logarithmic function. Here's a step-by-step guide:

• Start by isolating the squared term: \( -x^2 + 16 > 0 \) simplifies to \( 16 > x^2 \).
• Take the square root of both sides to solve for 'x': \( x^2 < 16 \) becomes \(-4 < x < 4\).

This solution indicates that 'x' must be between -4 and 4, meaning any value within this range will satisfy the inequality. This type of problem is foundational in understanding where functions can be defined, as it helps determine the domain.

Interval notation

Interval notation is a mathematical shorthand used to describe a set of numbers along a continuum. It efficiently conveys the span of numbers that form the domain of a function or the solution to an inequality.

When utilizing interval notation:

• Parentheses \(( )\) indicate that the endpoints are not included in the set (open interval).
• Brackets \([ ]\) indicate that the endpoints are included (closed interval).

In our example of finding the domain for \( f(x) = \ln(-x^2 + 16) \), after solving \(-x^2 + 16 > 0\), we determined that valid 'x' values fall between -4 and 4 but do not include -4 or 4 themselves. Therefore, in interval notation, this domain is expressed as \((-4, 4)\).
This notation succinctly represents all numbers greater than -4 and less than 4, making it an efficient way to communicate mathematical solutions.