Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, state the domain and range of the function. \(f(x)=\log _{2}(12-3 x)-3\)

Step-by-Step Solution

Answer

Domain: \((-\infty, 4)\); Range: \(( -\infty, \infty )\).

Step 1: Understanding the Function

The given function is \( f(x) = \log_{2}(12 - 3x) - 3 \). It is a logarithmic function. Logarithmic functions are only defined for positive arguments.

Step 2: Determine the Domain

For the function \( f(x) = \log_{2}(12 - 3x) - 3 \) to be defined, \( 12 - 3x > 0 \) must hold. Solve the inequality: \[ 12 - 3x > 0 \]\[ -3x > -12 \]\[ x < 4 \]So the domain is \( x < 4 \), or in interval notation, \((-\infty, 4)\).

Step 3: Determine the Range

A logarithmic function with the form \( \log_{b}(x) \) can result in any real number, where \( x > 0 \). Since the range of \( \log_{2}(12 - 3x) \) is all real numbers and subtracting 3 does not affect the range's extent, the overall range of \( f(x) \) is all real numbers, \(( -\infty, \infty )\).

Key Concepts

Logarithmic Functions

Logarithmic functions form the basis of our problem involving the expression \( f(x) = \log_{2}(12 - 3x) - 3 \). In general, a logarithmic function with base \( b \) is expressed as \( \log_{b}(x) \), where \( x \) must be positive. This constraint is because you cannot take the logarithm of a zero or negative number, as the function is not defined there.

A key aspect of logarithmic functions is their inverse relationship with exponential functions. This relationship helps solve equations and analyze function behavior.

• For the function to "exist," the inside part (known as the argument) must always be greater than zero.
• In our given function, \( \log_{2}(12-3x) \) implies the expression \( 12 - 3x > 0 \) must hold.

Since the base of our logarithm is 2, it is essential when graphing or solving these functions to remember that common bases are often 2 or 10 and each base focuses on different aspects of growth and decay.

Inequalities

Inequalities arise naturally when solving for the domain of logarithmic functions, ensuring that their arguments are positive. In our exercise, we set up this inequality:

• \( 12 - 3x > 0 \)
• Solving it, we subtract 12 from each side to get \( -3x > -12 \).
• Dividing by -3, we must flip the inequality sign, yielding \( x < 4 \).

This step is crucial, as flipping the inequality is often overlooked by students. It is an essential rule stemming from multiplying or dividing both sides of an inequality by a negative number.

Mastering inequalities will help in determining the range for various functions or when working with constraints in real-world applications. Always be mindful of the rules around changing inequality directions during manipulation.

Interval Notation

Interval notation is a powerful tool for conveying the domain and range of functions concisely. In mathematics, it is used to specify which numbers satisfy inequalities, and it especially shines when dealing with continuous sets of numbers.

• The domain for our function was found through solving \( x < 4 \), thus in interval notation, this becomes \((-\infty, 4)\).
• The range, unaffected by the expression \( -3 \), remains each real number, expressed as \((-\infty, \infty)\).

Using interval notation:

- Parentheses \(( )\) indicate that an endpoint is not included, as with \( 4 \), because it would make the argument zero, which is not defined.
- Brackets \([ ]\) indicate inclusion, such as when specifying a closed interval.

Mastery of interval notation can help students succinctly express solutions across a range of mathematical topics, ensuring clarity in communication.