To determine the domain of the function [tex]\( f(x) = 4^x \)[/tex], let's analyze the function step-by-step.
1. Understanding the function:
- The given function is [tex]\( f(x) = 4^x \)[/tex].
- In this function, [tex]\( 4 \)[/tex] is the base and [tex]\( x \)[/tex] is the exponent.
2. Exponential functions:
- Exponential functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a \)[/tex] is a positive constant) are defined for all real numbers [tex]\( x \)[/tex].
- There are no restrictions on [tex]\( x \)[/tex] for [tex]\( 4^x \)[/tex]. You can plug in any real number into the function, and it will yield a valid output.
3. Checking different types of [tex]\( x \)[/tex]:
- If [tex]\( x \)[/tex] is zero, [tex]\( f(x) = 4^0 = 1 \)[/tex].
- If [tex]\( x \)[/tex] is positive, for instance [tex]\( x = 2 \)[/tex], [tex]\( f(x) = 4^2 = 16 \)[/tex].
- If [tex]\( x \)[/tex] is negative, for instance [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 4^{-1} = \frac{1}{4} \)[/tex].
These examples show that [tex]\( f(x) = 4^x \)[/tex] produces a valid output for any real number [tex]\( x \)[/tex].
4. Conclusion:
- There are no values of [tex]\( x \)[/tex] that make [tex]\( 4^x \)[/tex] undefined. Thus, the function is defined for all real numbers.
Therefore, the domain of [tex]\( f(x) = 4^x \)[/tex] is all real numbers.
So, the correct answer is:
A. All real numbers
