Answer :
To determine the range of the function \( f(x) = \frac{2}{3}(6)^x \) after it has been reflected over the \( x \)-axis, let's go through the steps methodically.
1. Original Function Analysis:
- The original function is \( f(x) = \frac{2}{3}(6)^x \).
- For any exponent \( x \), \( 6^x \) is always positive since the base 6 is a positive number.
- Multiplying this positive value by \(\frac{2}{3}\), a positive number, means \( f(x) \) is always positive.
- Hence, the range of the original function \( f(x) \) is all real numbers greater than 0.
2. Reflecting Over the \( x \)-Axis:
- Reflecting a function over the \( x \)-axis involves changing the sign of the function's output.
- The transformed function, reflecting \( f(x) \) over the \( x \)-axis, becomes \( f'(x) = -\frac{2}{3}(6)^x \).
3. Range of the Reflected Function:
- In the reflected function \( f'(x) = -\frac{2}{3}(6)^x \), for each \( x \), the output \( \frac{2}{3}(6)^x \) is always positive, as discussed before.
- By multiplying this positive output by -1 (due to reflection), we switch the sign of the original outputs, making all outputs negative.
- Therefore, the function \( f'(x) \) yields only negative values.
4. Conclusion:
- The new range of the transformed function \( f'(x) \) is all real numbers less than or equal to 0 since it includes all negative values, and zero itself is attainable as an upper bound when considering the asymptotic behavior as \( x \to -\infty \).
Hence, the best description of the range of the function \( f(x) = \frac{2}{3}(6)^x \) after it has been reflected over the \( x \)-axis is:
all real numbers less than or equal to 0.
1. Original Function Analysis:
- The original function is \( f(x) = \frac{2}{3}(6)^x \).
- For any exponent \( x \), \( 6^x \) is always positive since the base 6 is a positive number.
- Multiplying this positive value by \(\frac{2}{3}\), a positive number, means \( f(x) \) is always positive.
- Hence, the range of the original function \( f(x) \) is all real numbers greater than 0.
2. Reflecting Over the \( x \)-Axis:
- Reflecting a function over the \( x \)-axis involves changing the sign of the function's output.
- The transformed function, reflecting \( f(x) \) over the \( x \)-axis, becomes \( f'(x) = -\frac{2}{3}(6)^x \).
3. Range of the Reflected Function:
- In the reflected function \( f'(x) = -\frac{2}{3}(6)^x \), for each \( x \), the output \( \frac{2}{3}(6)^x \) is always positive, as discussed before.
- By multiplying this positive output by -1 (due to reflection), we switch the sign of the original outputs, making all outputs negative.
- Therefore, the function \( f'(x) \) yields only negative values.
4. Conclusion:
- The new range of the transformed function \( f'(x) \) is all real numbers less than or equal to 0 since it includes all negative values, and zero itself is attainable as an upper bound when considering the asymptotic behavior as \( x \to -\infty \).
Hence, the best description of the range of the function \( f(x) = \frac{2}{3}(6)^x \) after it has been reflected over the \( x \)-axis is:
all real numbers less than or equal to 0.
