Problem 16 - College Algebra with Modeling and Visualization

Question

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log _{4}\left(4-x^{2}\right)$$

Step-by-Step Solution

Answer

The domain of the function is \\((-2, 2)\\).

Step 1: Understand the Function

The function given is a logarithmic function, specifically \(f(x) = \log_{4}(4-x^2)\). The logarithmic function is undefined for zero and negative arguments.

Step 2: Set the Argument Greater Than Zero

The expression inside the logarithm, which is \(4-x^2\), must be greater than zero: \[4-x^2 > 0.\]

Step 3: Solve the Inequality

Solve the inequality from Step 2. First, rewrite it as: \[x^2 < 4\] This can be rewritten as: \[-2 < x < 2\]

Step 4: Write the Domain in Interval Notation

The solution \(-2 < x < 2\) represents the set of all values that \(x\) can take. In interval notation, it is written as: \((-2, 2)\)

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse operations of exponential functions. They are used to determine the power to which a base number must be raised to obtain a given value. In the given problem, the function is expressed as:- \( f(x) = \log_4(4-x^2) \) where 4 is the base.It's important to remember:

• Logarithms are undefined for 0 and negative numbers, meaning the input or argument (inside the log) must be greater than zero.
• They transform multiplication into addition, aiding in complex calculations.

This understanding is crucial when solving for the domain of any logarithmic function, as all possible inputs must result in positive outputs before computations can proceed.

Inequalities

Inequalities express relationships between values or expressions that are not equal, using symbols: \( >, <, \geq, \leq \). In finding the domain of a logarithmic function, inequalities help to determine which values of \(x\) make the argument positive.The original exercise illustrated an inequality:

• \(4 - x^2 > 0\)

To solve this:

• Rearrange it to compare \(x^2\) with 4, yielding \(x^2 < 4\).
• Solve this by finding the square roots to get the range of \(x\): \(-2 < x < 2\).

This step is vital as it directly relates to determining which inputs are permissible for the logarithmic function. Inequality solutions should always check if the equalities align with the function's restrictions, especially in cases involving squares.

Interval Notation

Interval notation offers a concise way to represent a range of numbers. It uses brackets to demonstrate whether endpoints are included or not:

• Open brackets \(( \) and \() \) indicate endpoints are not included.
• Closed brackets \([ \) and \(] \) indicate endpoints are included.

For the problem given:

• The inequality \(-2 < x < 2\) is represented in interval notation as \((-2, 2)\).

Here, open brackets show that neither \(-2\) nor \(2\) are part of the domain, aligning with the requirement of the logarithm having a positive argument. Using interval notation is beneficial in mathematics as it provides clarity and a standardized shorthand for expressing continuous ranges of numbers.