Sure, let's work through the problem step-by-step.
First, let's understand the original function [tex]\( f(x) = 6 \)[/tex].
### Step 1: Analyze Original Function
The function [tex]\( f(x) = 6 \)[/tex] is a constant function. This means that for any input [tex]\( x \)[/tex], the output is always [tex]\( 6 \)[/tex]. Therefore, the range of the original function is simply:
[tex]\[ \{6\} \][/tex]
### Step 2: Reflect over the x-axis
Reflecting a function over the x-axis entails multiplying the output of the function by [tex]\(-1\)[/tex]. Therefore, our new function after reflecting over the x-axis is:
[tex]\[ g(x) = -f(x) = -6 \][/tex]
Thus, no matter the input [tex]\( x \)[/tex], the output is always [tex]\(-6\)[/tex].
### Step 3: Determine the Range of the Reflected Function
The range of the new function [tex]\( g(x) = -6 \)[/tex] is:
[tex]\[ \{-6\} \][/tex]
### Step 4: Analyze the Range of the Reflected Function
The range [tex]\( \{-6\} \)[/tex] signifies that the only output value is [tex]\(-6\)[/tex].
### Step 5: Compare with the Multiple Choices
Looking at the provided options:
- all real numbers
- all real numbers less than 0
- all real numbers greater than 0
- all real numbers less than or equal to 0
Since [tex]\(-6\)[/tex] is less than or equal to [tex]\( 0 \)[/tex], the correct description is:
[tex]\[ \text{all real numbers less than or equal to 0} \][/tex]
Thus, the best description of the range of the function after reflection over the x-axis is:
[tex]\[ \text{all real numbers less than or equal to 0} \][/tex]
This corresponds to the answer:
[tex]\[ 4 \][/tex]
