To determine the domain of the function [tex]\( f(x) = 3^{x-2} \)[/tex], we need to consider the nature of the function. This function is an exponential function, specifically of the form [tex]\( 3^{x-2} \)[/tex].
Exponential functions have certain characteristics:
1. The base of the exponent (which is 3 in this case) is a positive real number.
2. The exponent (which is [tex]\( x-2 \)[/tex] here) can be any real number.
Exponential functions are defined for all real numbers in their exponent because:
- The base (3 in our case) is always positive.
- Raising a positive number to any real exponent (positive, negative, or zero) will always result in a defined, real number.
Therefore:
- There is no restriction on [tex]\( x \)[/tex] that makes the function undefined.
- The exponent [tex]\( x-2 \)[/tex] can take any real value.
Hence, the domain of [tex]\( f(x) = 3^{x-2} \)[/tex] is the set of all real numbers.
So, the domain of [tex]\( f(x) = 3^{x-2} \)[/tex] is:
[tex]\[
\{x \mid x \text{ is a real number}\}
\][/tex]
This matches the answer provided. Thus, the domain of [tex]\( f(x) = 3^{x-2} \)[/tex] is:
[tex]\[
\{x \mid x \text{ is a real number}\}
\][/tex]
