Answer :
To analyze the function [tex]\( f(x) = 3 \left( \frac{1}{3} \right)^x \)[/tex], letâs consider each given statement one by one.
1. Each successive output is the previous output divided by 3.
- This statement is true. Given the function [tex]$ f(x) = 3 \left( \frac{1}{3} \right)^x $[/tex], the output for [tex]$ x + 1 $[/tex] can be written as [tex]$ f(x+1) = 3 \left( \frac{1}{3} \right)^{x+1} = 3 \left( \frac{1}{3} \right)^x \cdot \frac{1}{3} = \frac{1}{3} \cdot f(x) $[/tex]. Hence, indeed each successive output is the previous output divided by 3.
2. As the domain values increase, the range values decrease.
- This statement is true. Because [tex]\( \left( \frac{1}{3} \right)^x \)[/tex] decreases as [tex]$x$[/tex] increases, multiplying by 3 still results in a decreasing function.
3. The graph of the function is linear, decreasing from left to right.
- This statement is false. The function [tex]\( f(x) = 3 \left( \frac{1}{3} \right)^x \)[/tex] is an exponential function, not a linear one. Therefore, its graph is not a straight line but rather an exponentially decaying curve.
4. Each successive output is the previous output multiplied by 3.
- This statement is false. We have already shown in the first statement that each successive output is the previous output divided by 3, not multiplied by 3.
5. The range of the function is all real numbers greater than 0.
- This statement is true. Since [tex]\( f(x) = 3 \left( \frac{1}{3} \right)^x \)[/tex] is always positive for all real numbers [tex]$x$[/tex], and can take any positive value as [tex]$x$[/tex] varies, the range is all real numbers greater than 0.
6. The domain of the function is all real numbers greater than 0.
- This statement is false. The domain of the function [tex]\( f(x) = 3 \left( \frac{1}{3} \right)^x \)[/tex] is all real numbers. There is no restriction on the values that [tex]$x$[/tex] can take.
The correct responses are:
- Each successive output is the previous output divided by 3. (True)
- As the domain values increase, the range values decrease. (True)
- The graph of the function is linear, decreasing from left to right. (False)
- Each successive output is the previous output multiplied by 3. (False)
- The range of the function is all real numbers greater than 0. (True)
- The domain of the function is all real numbers greater than 0. (False)
Thus, the correct results align with:
```
(True, True, False, False, True, False)
```
1. Each successive output is the previous output divided by 3.
- This statement is true. Given the function [tex]$ f(x) = 3 \left( \frac{1}{3} \right)^x $[/tex], the output for [tex]$ x + 1 $[/tex] can be written as [tex]$ f(x+1) = 3 \left( \frac{1}{3} \right)^{x+1} = 3 \left( \frac{1}{3} \right)^x \cdot \frac{1}{3} = \frac{1}{3} \cdot f(x) $[/tex]. Hence, indeed each successive output is the previous output divided by 3.
2. As the domain values increase, the range values decrease.
- This statement is true. Because [tex]\( \left( \frac{1}{3} \right)^x \)[/tex] decreases as [tex]$x$[/tex] increases, multiplying by 3 still results in a decreasing function.
3. The graph of the function is linear, decreasing from left to right.
- This statement is false. The function [tex]\( f(x) = 3 \left( \frac{1}{3} \right)^x \)[/tex] is an exponential function, not a linear one. Therefore, its graph is not a straight line but rather an exponentially decaying curve.
4. Each successive output is the previous output multiplied by 3.
- This statement is false. We have already shown in the first statement that each successive output is the previous output divided by 3, not multiplied by 3.
5. The range of the function is all real numbers greater than 0.
- This statement is true. Since [tex]\( f(x) = 3 \left( \frac{1}{3} \right)^x \)[/tex] is always positive for all real numbers [tex]$x$[/tex], and can take any positive value as [tex]$x$[/tex] varies, the range is all real numbers greater than 0.
6. The domain of the function is all real numbers greater than 0.
- This statement is false. The domain of the function [tex]\( f(x) = 3 \left( \frac{1}{3} \right)^x \)[/tex] is all real numbers. There is no restriction on the values that [tex]$x$[/tex] can take.
The correct responses are:
- Each successive output is the previous output divided by 3. (True)
- As the domain values increase, the range values decrease. (True)
- The graph of the function is linear, decreasing from left to right. (False)
- Each successive output is the previous output multiplied by 3. (False)
- The range of the function is all real numbers greater than 0. (True)
- The domain of the function is all real numbers greater than 0. (False)
Thus, the correct results align with:
```
(True, True, False, False, True, False)
```
