Problem 39 - Precalculus with Calculus Previews

Question

In Problems \(39-44\), find the domain of the given function \(f\). $$ f(x)=\ln (2 x-3) $$

Step-by-Step Solution

Answer

The domain of the function is \((\frac{3}{2}, \infty)\).

Step 1: Understand the Function and Domain Constraints

The function given is a logarithmic function: \(f(x) = \ln(2x - 3)\). The domain of a logarithmic function, \(\ln(u)\), is the set of values for which \(u > 0\). So, for \(f(x)\), we need \(2x - 3 > 0\).

Step 2: Solve the Inequality for x

To find the domain, solve the inequality \(2x - 3 > 0\). Add 3 to both sides to get \(2x > 3\). Then, divide both sides by 2 to isolate \(x\):\[x > \frac{3}{2}\].

Step 3: State the Domain of the Function

The solution to the inequality \(x > \frac{3}{2}\) gives us the domain of the function. The domain of \(f(x) = \ln(2x - 3)\) is all real numbers greater than \(\frac{3}{2}\), which can be expressed in interval notation as \((\frac{3}{2}, \infty)\).

Key Concepts

Logarithmic Functions

Logarithmic functions are a type of mathematical function used to solve equations involving logs or systems that grow or decay exponentially. An example of a basic logarithmic function can be written as \(f(x) = \ln(x)\). Here, the function is the natural logarithm of \(x\), which is the opposite of an exponential function. The property you should remember is that logarithms are only defined for positive inputs.

• This means that the argument inside the logarithm (replacing the \(x\) in our example) must be greater than zero.
• For any logarithmic function \(\ln(u)\), it is crucial to first identify the domain by ensuring that \(u > 0\).

Logarithmic functions are widely used in various scientific fields because they can model real-world scenarios like population growth, radioactive decay, and the pH scale in chemistry. These applications arise because they offer a straightforward way to describe these non-linear relationships.

Inequalities

Inequalities are mathematical statements that relate expressions that may not be equal. They are often used to express conditions or constraints in algebra and other areas of mathematics. When you solve inequalities, you determine the range of values that satisfy the given inequality.

In the case of our problem, we want to solve the inequality to find when the log function is defined, i.e., when \(2x - 3 > 0\). Solving inequalities typically involves using the same rules as solving equations:

• First, add or subtract terms to both sides to simplify the expression.
• Then, multiply or divide by a constant to isolate the variable, always being careful to reverse the inequality sign if multiplying or dividing by a negative number.

Once the solution is found, it may be useful to verify it by substituting values from the resulting range back into the original inequality to ensure they satisfy the conditions stated.

Interval Notation

Interval notation is a shorthand used in mathematics to express the set of numbers within a specified range. It's a popular way to describe domains and solutions to inequalities.

Consider the inequality solution \(x > \frac{3}{2}\). In interval notation, this is expressed as \(\left(\frac{3}{2}, \infty\right)\). Here's what the notation means:

• The parenthesis \(\left(\right)\) indicates that the endpoint is not included, which is why it's used for a condition like \(x > \frac{3}{2}\) (not including \(\frac{3}{2}\)).
• The infinity symbol (\(\infty\)) shows that the set continues indefinitely.

Interval notation is concise and eliminates the need for using inequality symbols, making it a preferred method for representing solutions in mathematics. By understanding how to apply it, students can effectively communicate their results and interpret the results of others.