Let's consider the function given: [tex]\( f(x) = \frac{2}{3} \cdot 6^x \)[/tex].
1. Understanding the Original Function:
- The function [tex]\( 6^x \)[/tex] is an exponential function. For any real number [tex]\( x \)[/tex], the value of [tex]\( 6^x \)[/tex] is always positive.
- Since [tex]\( \frac{2}{3} \)[/tex] is a constant positive multiplier, multiplying [tex]\( 6^x \)[/tex] by [tex]\( \frac{2}{3} \)[/tex] does not change the fact that the function is always positive.
- Therefore, the range of [tex]\( f(x) = \frac{2}{3} \cdot 6^x \)[/tex] is all real numbers greater than 0.
2. Reflecting Over the x-axis:
- Reflecting a function over the x-axis involves multiplying the function by [tex]\(-1\)[/tex].
- The new function after reflection is [tex]\( g(x) = - \left( \frac{2}{3} \cdot 6^x \right) \)[/tex].
3. Determining the Range of the Reflected Function:
- Since [tex]\( \frac{2}{3} \cdot 6^x \)[/tex] is always positive for all real [tex]\( x \)[/tex], multiplying by [tex]\(-1\)[/tex] will make these values always negative (or zero if there is a constant term involved, but here there is not).
- Thus, [tex]\( g(x) = - \left( \frac{2}{3} \cdot 6^x \right) \)[/tex] will produce all negative real numbers as output.
- Therefore, the range of [tex]\( g(x) = - \left( \frac{2}{3} \cdot 6^x \right) \)[/tex] is all real numbers less than or equal to 0.
Based on this reasoning, the range of the function [tex]\( f(x) = \frac{2}{3} \cdot 6^x \)[/tex] after it has been reflected over the x-axis is best described as:
all real numbers less than or equal to 0.
