Answer :
To determine which statements correctly describe the function \(f(x) = 3\left(\frac{1}{3}\right)^x\), let's analyze each statement one by one.
1. Each successive output is the previous output divided by 3.
- Let's consider the functional form of \(f(x)\):
[tex]\[ f(x) = 3 \left(\frac{1}{3}\right)^x \][/tex]
If you calculate \(f(x+1)\):
[tex]\[ f(x+1) = 3 \left(\frac{1}{3}\right)^{x+1} = 3 \left(\frac{1}{3}\right)^x \cdot \frac{1}{3} = f(x) \cdot \frac{1}{3} \][/tex]
This shows that each successive output is indeed the previous output divided by 3. Hence, this statement is true.
2. As the domain values increase, the range values decrease.
- Observing the form of the function again:
[tex]\[ f(x) = 3 \left(\frac{1}{3}\right)^x \][/tex]
Since \(\left(\frac{1}{3}\right)^x\) is a decreasing function for increasing values of \(x\), it means as \(x\) (the domain) increases, \(f(x)\) (the range) decreases. Hence, this statement is true.
3. The graph of the function is linear, decreasing from left to right.
- The function \(f(x) = 3\left(\frac{1}{3}\right)^x\) is an exponential function, not a linear function. Exponential functions do not produce straight-line graphs. Therefore, this statement is false.
4. Each successive output is the previous output multiplied by 3.
- From the earlier calculation, we observed:
[tex]\[ f(x+1) = f(x) \times \frac{1}{3} \][/tex]
Hence, each successive output is the previous output divided by 3, not multiplied by 3. Therefore, this statement is false.
5. The range of the function is all real numbers greater than 0.
- Analyzing the behavior of \(f(x)\):
[tex]\[ f(x) = 3\left(\frac{1}{3}\right)^x \][/tex]
The base \(\left(\frac{1}{3}\right)^x\) is always positive (but never zero) for any real \(x\). Since it is scaled by 3, \(f(x)\) will also always be positive but never zero, ranging from \(3\) to values approaching \(0\) but never reaching zero. Therefore, the range is all real numbers greater than 0. Hence, this statement is true.
6. The domain of the function is all real numbers greater than 0.
- The domain of the function refers to all the possible input values \(x\). Since \(f(x) = 3\left(\frac{1}{3}\right)^x\) is well-defined for any real number \(x\), the domain is all real numbers, not limited to values greater than zero. Therefore, this statement is false.
Thus, the correct statements are:
- Each successive output is the previous output divided by 3.
- As the domain values increase, the range values decrease.
- The range of the function is all real numbers greater than 0.
These correspond to statements [1, 2, 5].
1. Each successive output is the previous output divided by 3.
- Let's consider the functional form of \(f(x)\):
[tex]\[ f(x) = 3 \left(\frac{1}{3}\right)^x \][/tex]
If you calculate \(f(x+1)\):
[tex]\[ f(x+1) = 3 \left(\frac{1}{3}\right)^{x+1} = 3 \left(\frac{1}{3}\right)^x \cdot \frac{1}{3} = f(x) \cdot \frac{1}{3} \][/tex]
This shows that each successive output is indeed the previous output divided by 3. Hence, this statement is true.
2. As the domain values increase, the range values decrease.
- Observing the form of the function again:
[tex]\[ f(x) = 3 \left(\frac{1}{3}\right)^x \][/tex]
Since \(\left(\frac{1}{3}\right)^x\) is a decreasing function for increasing values of \(x\), it means as \(x\) (the domain) increases, \(f(x)\) (the range) decreases. Hence, this statement is true.
3. The graph of the function is linear, decreasing from left to right.
- The function \(f(x) = 3\left(\frac{1}{3}\right)^x\) is an exponential function, not a linear function. Exponential functions do not produce straight-line graphs. Therefore, this statement is false.
4. Each successive output is the previous output multiplied by 3.
- From the earlier calculation, we observed:
[tex]\[ f(x+1) = f(x) \times \frac{1}{3} \][/tex]
Hence, each successive output is the previous output divided by 3, not multiplied by 3. Therefore, this statement is false.
5. The range of the function is all real numbers greater than 0.
- Analyzing the behavior of \(f(x)\):
[tex]\[ f(x) = 3\left(\frac{1}{3}\right)^x \][/tex]
The base \(\left(\frac{1}{3}\right)^x\) is always positive (but never zero) for any real \(x\). Since it is scaled by 3, \(f(x)\) will also always be positive but never zero, ranging from \(3\) to values approaching \(0\) but never reaching zero. Therefore, the range is all real numbers greater than 0. Hence, this statement is true.
6. The domain of the function is all real numbers greater than 0.
- The domain of the function refers to all the possible input values \(x\). Since \(f(x) = 3\left(\frac{1}{3}\right)^x\) is well-defined for any real number \(x\), the domain is all real numbers, not limited to values greater than zero. Therefore, this statement is false.
Thus, the correct statements are:
- Each successive output is the previous output divided by 3.
- As the domain values increase, the range values decrease.
- The range of the function is all real numbers greater than 0.
These correspond to statements [1, 2, 5].
