Problem 13 - College Algebra with Modeling and Visualization

Question

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log (x+3)$$

Step-by-Step Solution

Answer

The domain of the function is \((-3, \infty)\) in interval notation.

Step 1: Understand Logarithmic Function Domain

For the function \(f(x) = \log(x+3)\), the expression inside the logarithm, \(x + 3\), must be positive. This is the primary condition that defines the domain of logarithmic functions: \(x+3 > 0\).

Step 2: Solve for \(x\) to Find Domain

To find the values of \(x\) that satisfy \(x + 3 > 0\), solve the inequality. Subtract 3 from both sides: \(x > -3\).

Step 3: Express Domain in Interval Notation

The solution \(x > -3\) implies that \(x\) can be any real number greater than \(-3\). Express this in interval notation as \((-3, \infty)\).

Step 4: Express Domain in Set Builder Notation

Alternatively, express the domain using set-builder notation: \(\{x \in \mathbb{R} \mid x > -3\}\).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions that involve the logarithm of a variable. In simpler terms, the logarithm asks the question "To what power must a certain base be raised, in order to yield a given number?" A common logarithmic function is written in the form \(f(x) = \log_b(x)\), where \(b\) is the base of the logarithm. One key detail to remember is that the logarithm is only defined for positive numbers.
This means the expression inside the logarithm must be greater than zero. For instance, in the problem provided where \(f(x) = \log(x + 3)\), you must ensure \(x + 3 > 0\).
Solving this inequality helps determine the range of \(x\) values, ensuring that the expression inside the logarithm remains positive. If this condition is met, the logarithmic function will yield real outputs, making the function valid and operable.

Setbuilder Notation

Setbuilder notation is a mathematical shorthand used to define a set by specifying a property that its members must satisfy. It's typically written using curly braces and includes a condition that elements of the set must fulfill.
For example, when defining the domain of a function using setbuilder notation, the format is generally \(\{x \in A \mid \text{condition}\}\). Here, \(A\) usually represents the set of all possible numbers, such as the real numbers \(\mathbb{R}\).
In the case of the logarithmic function \(f(x) = \log(x + 3)\), where we determined that the domain inequality is \(x > -3\), the setbuilder notation would be written as \(\{x \in \mathbb{R} \mid x > -3\}\).
This concise notation quickly summarizes the condition that defines the domain of the function.

Interval Notation

Interval notation provides a way of writing the range of values that a variable can take, using intervals. It is a concise way to represent continuous subsets of real numbers. Intervals can be open, closed, or half-open/closed, depending on whether the endpoints are included in the set.
An open interval does not include its endpoints, indicated by parentheses like \((a, b)\). A closed interval includes its endpoints, shown with square brackets, \([a, b]\).
For the expression \(x > -3\), derived from \(f(x) = \log(x + 3)\), we use an open interval to express values greater than \(-3\). This is noted as \((-3, \infty)\), meaning that \(x\) can be any value greater than \(-3\).
This notation provides a clear and efficient way to communicate the allowable values for \(x\) in the domain of a function.