IDNLearn.com is your go-to resource for finding answers to any question you have. Join our interactive community and access reliable, detailed answers from experienced professionals across a variety of topics.
Sagot :
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.
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.
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Your search for solutions ends at IDNLearn.com. Thank you for visiting, and we look forward to helping you again.