Let's analyze the functions [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex] and [tex]\( g(x) = \log_{\frac{1}{3}} x \)[/tex] to determine which statement is true.
### Step 1: Identify the Domain and Range of [tex]\( f(x) \)[/tex]
For the function [tex]\( f(x) = \left( \frac{1}{3} \right)^x \)[/tex]:
- The domain of [tex]\( f(x) \)[/tex] is all real numbers, [tex]\( \mathbb{R} \)[/tex], since you can raise a positive number to any real power.
- The range of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex]. This is because an exponential function with a positive base (but less than 1) will always produce positive values, but never zero or negative values.
### Step 2: Identify the Domain and Range of [tex]\( g(x) \)[/tex]
For the function [tex]\( g(x) = \log_{\frac{1}{3}} x \)[/tex]:
- The domain of [tex]\( g(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex]. This is because the logarithm is only defined for positive real numbers.
- The range of [tex]\( g(x) \)[/tex] is all real numbers, [tex]\( \mathbb{R} \)[/tex]. This is because the logarithm function (with any positive base not equal to 1) can produce any real number as an output.
### Step 3: Compare the Options
Let's evaluate each option based on our findings:
- Option A: Domain of [tex]\( f(x) \)[/tex] = Range of [tex]\( g(x) = \mathbb{R} \)[/tex]
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( \mathbb{R} \)[/tex], and the range of [tex]\( g(x) \)[/tex] is [tex]\( \mathbb{R} \)[/tex]. So, this statement is true.
- Option B: Range of [tex]\( f(x) \)[/tex] = Domain of [tex]\( g(x) = \mathbb{R} \)[/tex]
- The range of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex], and the domain of [tex]\( g(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex]. So, this statement is not true.
- Option C: Domain of [tex]\( g(x) \)[/tex] = Range of [tex]\( f(x) = \mathbb{R} \)[/tex]
- The domain of [tex]\( g(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex], and the range of [tex]\( f(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex]. So, this statement is not true.
- Option D: Range of [tex]\( g(x) \)[/tex] = Domain of [tex]\( f(x) = \mathbb{R}^+ \)[/tex]
- The range of [tex]\( g(x) \)[/tex] is [tex]\( \mathbb{R} \)[/tex], and the domain of [tex]\( f(x) \)[/tex] is [tex]\( \mathbb{R} \)[/tex]. So, this statement is not true.
Among all the options, Option A is correct.
Thus, the correct answer is:
[tex]\[ \boxed{21} \][/tex]
