Determining the Domain of an Exponential Function \( f(x) = 3^x \)

The question asks for the domain of the function \( f(x) = 3^x \). The domain of a function is the set of all possible input values (x-values) for which the function is defined in the real number system.

Analyzing the Function \( f(x) = 3^x \)

The function \( f(x) = 3^x \) is an exponential function. It has the form \( f(x) = a^x \), where the base \(a\) is a positive real number and \(a \neq 1\). In this specific function, the base \(a\) is 3.

Domain of Exponential Functions

For exponential functions of the form \( f(x) = a^x \), where the base \(a\) is a positive real number (\(a > 0\)) and not equal to 1 (\(a \neq 1\)), the function is defined for all real numbers \(x\). There are no restrictions on the values that \(x\) can take in the exponent. You can raise a positive base to any real power (positive, negative, or zero).

• If \(x\) is a positive integer, \(3^x\) means multiplying 3 by itself \(x\) times.
• If \(x\) is a negative integer, \(3^x = 1/3^{-x}\), which is well-defined.
• If \(x\) is zero, \(3^0 = 1\), which is well-defined.
• If \(x\) is a rational number \(p/q\), \(3^{p/q} = \sqrt[q]{3^p}\), which is well-defined for a positive base.
• If \(x\) is an irrational number, \(3^x\) is defined as the limit of \(3^r\) as rational numbers \(r\) approach \(x\).

Since there are no values of \(x\) that would make \( f(x) = 3^x \) undefined in the real number system, the domain is all real numbers.

Expressing the Domain in Interval Notation

The set of all real numbers is represented in interval notation as \( (-\infty, \infty) \).

Evaluating the Given Options

• Option 1: \( (-\infty, \infty) \) - This represents all real numbers. This matches our finding for the domain of \( f(x) = 3^x \).
• Option 2: \( (0, \infty) \) - This represents all positive real numbers. This is the range of \( f(x) = 3^x \), not the domain.
• Option 3: \( [0, \infty) \) - This represents all non-negative real numbers. This is also not the domain; the function is defined for negative \(x\) values.
• Option 4: \( (-\infty, \infty) - \{0\} \) - This represents all real numbers except 0. This is incorrect; the function is defined at \(x=0\) ( \(3^0 = 1\) ).

Based on the analysis, the domain of the function \( f(x) = 3^x \) is the set of all real numbers, which is \( (-\infty, \infty) \).

Revision Table: Understanding Function Domain

Additional Information: Exponential Functions

• An exponential function has the form \( f(x) = a^x \), where \(a\) is the base and \(x\) is the exponent.
• For the function to be a standard exponential function used in calculus and algebra, the base \(a\) must be positive (\(a > 0\)) and not equal to 1 (\(a \neq 1\)).
• If \(a=1\), \(f(x) = 1^x = 1\), which is a constant function, not typically considered exponential.
• If \(a \le 0\), \(a^x\) might not be defined for all real \(x\) (e.g., \( (-2)^{1/2} = \sqrt{-2} \) is not a real number).
• The range of \( f(x) = a^x \) with \(a > 0, a \neq 1\) is always \( (0, \infty) \) (all positive real numbers). The graph is always above the x-axis.