Problem 63 - Precalculus

Question

State the domain of the logarithmic function in interval notation. $$f(x)=\log \left(x^{2}+1\right)$$

Step-by-Step Solution

Answer

The domain of \(f(x) = \log(x^2 + 1)\) is \((-\infty, \infty)\).

Step 1: Understanding the Logarithmic Function

The function given is a logarithmic function: \(f(x) = \log(x^2 + 1)\). For a logarithmic function, the argument (\(x^2 + 1\) here) must be greater than zero.

Step 2: Set Inequality for the Argument of the Logarithm

Since the expression inside the logarithm must be positive, set the inequality: \(x^2 + 1 > 0\).

Step 3: Analyze the Inequality

Notice that \(x^2 + 1\) is the sum of a squared term and 1, which makes it positive for all real \(x\). A squared term \(x^2\) is always non-negative and adding 1 ensures it is always positive.

Step 4: Conclude the Domain

Since \(x^2 + 1 > 0\) for all real numbers \(x\), the domain is all real numbers.

Key Concepts

Inequalities in Logarithms

When dealing with logarithmic functions, understanding the inequalities that the arguments must satisfy is crucial. A logarithmic function is defined only when its argument is greater than zero. This is because logarithms are the inverses of exponential functions, and they only accept positive values as inputs. Hence, for any function of the form:

• \( f(x) = \log(g(x)) \)

To solve for the domain of such a function, you need to determine where the argument \(g(x)\) is greater than zero. In our example with \( f(x) = \log(x^2 + 1) \), we set the inequality: \(x^2 + 1 > 0\).
The crucial aspect here is solving inequalities involving quadratic expressions or other complex forms that define the argument. Here, analysis of \(x^2 + 1 > 0\) shows that, since \(x^2\) is always non-negative and adding 1 ensures a positive result, all values for \(x\) satisfy the inequality. Thus, the function is defined over all real \(x\).

Interval Notation

Interval notation is a simplified way of expressing a set of numbers, typically used to denote the domain and range of functions. It uses brackets and parentheses to indicate the set boundaries:

• Use a square bracket \([ \text{or} ]\) to include an endpoint in the set
• Use a parenthesis \(( \text{or} )\) to exclude an endpoint

For example, the interval \([a, b]\) includes all numbers from \(a\) to \(b\), whereas \((a, b]\) includes all numbers from \(a\) (not inclusive) to \(b\) (inclusive). Also, \([-\infty, +\infty]\) denotes all real numbers.
In the context of our function \( f(x) = \log(x^2 + 1) \), this translates into the domain being expressed in interval notation as \(( -\infty, +\infty )\) because every real number \(x\) results in a positive outcome for \(x^2 + 1\). Thus, the function's domain spans all real numbers without restriction.

Real Numbers Domain

The concept of the domain in real numbers primarily involves finding all possible input values \(x\) for which a given function is defined. In mathematics, the domain of a function includes all the real numbers that make the function's expression valid and calculable without leading to undefined operations.
For helping with this understanding, remember:

• The domain excludes any value of \(x\) that would lead to a zero or negative input for a logarithm, division by zero, or a negative square root.
• In logarithmic functions, the argument must strictly be positive, therefore substantially determining the domain.

In our case with the function \( f(x) = \log(x^2 + 1) \), the polynomial expression in the argument is \(x^2 + 1\), which is always positive for all real \(x\). Thus, when expressing the domain, we say it covers all real numbers, denoted as \(( -\infty, +\infty )\). This implies no restriction over the real number line, making any \(x\) value a valid input for the function.