Problem 16 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The domain is \((-4, 4)\).

Step 1: Understand the Logarithmic Function's Requirement

To find the domain of a logarithmic function, we must identify when the expression inside the logarithm is greater than zero. For the function \( y = \ln(-x^2 + 16) \), this means we need to solve where \( -x^2 + 16 > 0 \).

Step 2: Rearrange and Simplify the Inequality

Rearrange the inequality \( -x^2 + 16 > 0 \) to find suitable values for \( x \). This can be rewritten as \( 16 > x^2 \). By further simplifying, we can express this as \( x^2 < 16 \).

Step 3: Solve the Inequality

Solve the inequality \( x^2 < 16 \). This means \( x \) must be between the square roots of 16, without including them. Therefore, \( -4 < x < 4 \).

Step 4: Express the Domain Numerically

Based on the solved inequality, the domain of the function can be expressed as the interval \( (-4, 4) \), meaning \( x \) can take any real number between -4 and 4, not inclusive.

Step 5: Verify with a Graph (Optional)

Optionally, graph the function \( y = \ln(-x^2 + 16) \). The graph should be visible between \( x = -4 \) and \( x = 4 \), confirming these are the values that keep the logarithmic expression positive.

Key Concepts

Inequalities

Understanding inequalities is crucial when solving for the domain of logarithmic functions. Inequalities tell us the range of values a variable can take that satisfy a given condition. For the exercise at hand, we must solve the inequality \(-x^2 + 16 > 0\).

To simplify, first rearrange it to \(x^2 < 16\).

This means we are looking for values of \(x\) such that when squared, they are less than 16.

• Taking the square root on both sides yields two conditions: \(-4 < x < 4\).
• These values indicate that the expression inside the logarithm remains positive, keeping the function defined.

In summary, solving the inequality gives the domain in which the logarithmic function is valid.

Logarithmic Functions

A logarithmic function is defined only for positive inputs. Therefore, finding the domain involves making sure the argument (input) stays positive. In this exercise, the function is\(y = \ln(-x^2 + 16)\),

meaning we seek points where\(-x^2 + 16 > 0\).

Logarithms naturally apply to processes or phenomena that change exponentially, such as growth or decay.

• The expression must always be positive, as logarithms of zero or negative numbers are undefined.
• It's essential to simplify the function's argument to understand where it stays greater than zero.Using the natural logarithm, \(\ln(x)\), helps quantify relationships of rapid change.

Thus, handling logarithmic functions requires a careful treatment of their arguments.

Graphical Representation

Graphing a logarithmic function can substantially aid in verifying its domain. It visualizes where the function exists, highlighting the range of permissible \(x\) values. In our example, plotting \(y = \ln(-x^2 + 16)\)

displays a curve that only appears between \(-4 < x < 4\).

• Outside this interval, the plot would not exist due to the argument turning negative or zero.
• A graph inherently reveals this constraint by showing a gap outside the permissible range.

Graphical insights are particularly useful when verifying the conditions deduced algebraically. They confirm the analytical solutions and provide an intuitive understanding of the function's behavior.