Answer :
Let's analyze the given function [tex]\( f(x) = \frac{2}{3} (6)^x \)[/tex] and see what happens when it is reflected over the [tex]\( x \)[/tex]-axis.
### Step-by-Step Solution:
1. Understand the Original Function:
The function [tex]\( f(x) = \frac{2}{3} (6)^x \)[/tex] is an exponential function with base 6. Since the base is a positive number greater than 1, and it's being multiplied by a positive constant [tex]\(\frac{2}{3}\)[/tex], the function [tex]\( f(x) \)[/tex] will always yield positive values for any real [tex]\( x \)[/tex].
2. Behavior of the Original Function:
For any real number [tex]\( x \)[/tex], [tex]\( 6^x \)[/tex] is always positive. When you multiply a positive number ([tex]\( 6^x \)[/tex]) by another positive number ([tex]\( \frac{2}{3} \)[/tex]), the result remains positive. Therefore, the range of the original function [tex]\( f(x) = \frac{2}{3} (6)^x \)[/tex] is all positive real numbers.
3. Reflection Over the [tex]\( x \)[/tex]-Axis:
Reflecting the function [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis essentially means you take the negative of the original function:
[tex]\[ g(x) = -f(x) = -\left(\frac{2}{3} (6)^x\right) = -\frac{2}{3} (6)^x \][/tex]
When you reflect an exponential function over the [tex]\( x \)[/tex]-axis, each positive value of the original function becomes its negative counterpart.
4. Analyzing the Reflected Function:
After reflection, [tex]\( g(x) = -\frac{2}{3} (6)^x \)[/tex] will always be negative for any real number [tex]\( x \)[/tex]. This is because [tex]\( \frac{2}{3} (6)^x \)[/tex] is always positive, and multiplying it by -1 makes it always negative.
5. Determining the Range:
Since [tex]\( g(x) = -\frac{2}{3} (6)^x \)[/tex] takes all positive values from [tex]\( f(x) \)[/tex] and converts them to negative values, the range of [tex]\( g(x) \)[/tex] is all real numbers less than 0.
Therefore, the range of the function after it has been reflected over the [tex]\( x \)[/tex]-axis is best described as:
[tex]\[ \text{all real numbers less than 0} \][/tex]
So, the correct answer is:
### all real numbers less than 0
### Step-by-Step Solution:
1. Understand the Original Function:
The function [tex]\( f(x) = \frac{2}{3} (6)^x \)[/tex] is an exponential function with base 6. Since the base is a positive number greater than 1, and it's being multiplied by a positive constant [tex]\(\frac{2}{3}\)[/tex], the function [tex]\( f(x) \)[/tex] will always yield positive values for any real [tex]\( x \)[/tex].
2. Behavior of the Original Function:
For any real number [tex]\( x \)[/tex], [tex]\( 6^x \)[/tex] is always positive. When you multiply a positive number ([tex]\( 6^x \)[/tex]) by another positive number ([tex]\( \frac{2}{3} \)[/tex]), the result remains positive. Therefore, the range of the original function [tex]\( f(x) = \frac{2}{3} (6)^x \)[/tex] is all positive real numbers.
3. Reflection Over the [tex]\( x \)[/tex]-Axis:
Reflecting the function [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis essentially means you take the negative of the original function:
[tex]\[ g(x) = -f(x) = -\left(\frac{2}{3} (6)^x\right) = -\frac{2}{3} (6)^x \][/tex]
When you reflect an exponential function over the [tex]\( x \)[/tex]-axis, each positive value of the original function becomes its negative counterpart.
4. Analyzing the Reflected Function:
After reflection, [tex]\( g(x) = -\frac{2}{3} (6)^x \)[/tex] will always be negative for any real number [tex]\( x \)[/tex]. This is because [tex]\( \frac{2}{3} (6)^x \)[/tex] is always positive, and multiplying it by -1 makes it always negative.
5. Determining the Range:
Since [tex]\( g(x) = -\frac{2}{3} (6)^x \)[/tex] takes all positive values from [tex]\( f(x) \)[/tex] and converts them to negative values, the range of [tex]\( g(x) \)[/tex] is all real numbers less than 0.
Therefore, the range of the function after it has been reflected over the [tex]\( x \)[/tex]-axis is best described as:
[tex]\[ \text{all real numbers less than 0} \][/tex]
So, the correct answer is:
### all real numbers less than 0
