To determine the range of the function [tex]\( y = \left( \frac{1}{3} \right)^x \)[/tex], we'll analyze the behavior of the function as [tex]\( x \)[/tex] varies over all real numbers.
1. Behavior as [tex]\( x \)[/tex] Approaches Positive Infinity:
- As [tex]\( x \)[/tex] becomes very large positively, the exponent [tex]\( x \)[/tex] causes the base [tex]\( \frac{1}{3} \)[/tex] to be raised to a very large power. Since [tex]\( \frac{1}{3} \)[/tex] is a fraction less than 1, its large powers will result in values approaching 0, but never actually reaching 0.
- Therefore, as [tex]\( x \to +\infty \)[/tex], [tex]\( y \to 0^+ \)[/tex].
2. Behavior as [tex]\( x \)[/tex] Approaches Negative Infinity:
- As [tex]\( x \)[/tex] becomes very large negatively, the exponent [tex]\( x \)[/tex] causes the base [tex]\( \frac{1}{3} \)[/tex] to be raised to a large negative power. This is equivalent to raising 3 to a large positive power (since [tex]\( \left( \frac{1}{3} \right)^{-x} = 3^x \)[/tex]).
- Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( y \to +\infty \)[/tex].
3. Range:
- Combining the above behaviors, for [tex]\( x \)[/tex] taking any real value, the output [tex]\( y \)[/tex] varies from values arbitrarily close to 0 (but never reaching 0) to values that can be infinitely large. Thereby, [tex]\( y \)[/tex] can take any value greater than 0.
Thus, the range of the function [tex]\( y = \left( \frac{1}{3} \right)^x \)[/tex] is [tex]\( (0, \infty) \)[/tex]. This means [tex]\( y \)[/tex] can be any positive real number, but not zero or negative.
Given the options:
- [tex]\( R - \{ 3 \} \)[/tex]
- [tex]\( - R \)[/tex]
- [tex]\( R \)[/tex]
- [tex]\( + R \)[/tex]
The most appropriate option that represents [tex]\( (0, \infty) \)[/tex] is [tex]\( + R \)[/tex]. This corresponds to the set of all positive real numbers.
So, the range of [tex]\( y = \left( \frac{1}{3} \right)^x \)[/tex] is [tex]\( +R \)[/tex].
