Question
Which of the following patterns shows f(0) , f(1) , f(2) , f(3) for the function f(x) = 1000 x (1/2)^x?
Responses
A 1000, 500, 250, 1251000, 500, 250, 125
B 1000, 2000, 4000, 80001000, 2000, 4000, 8000
C 1000, 999.5, 999, 998.51000, 999.5, 999, 998.5
D 1000, 1000.5, 1001, 1001.5
Responses
A 1000, 500, 250, 1251000, 500, 250, 125
B 1000, 2000, 4000, 80001000, 2000, 4000, 8000
C 1000, 999.5, 999, 998.51000, 999.5, 999, 998.5
D 1000, 1000.5, 1001, 1001.5
Answers
GPT-4o mini
To find the values of \( f(0) \), \( f(1) \), \( f(2) \), and \( f(3) \) for the function \( f(x) = 1000 x \left( \frac{1}{2} \right)^x \), we can calculate each one step by step.
1. **Calculate \( f(0) \)**:
\[
f(0) = 1000 \cdot 0 \cdot \left( \frac{1}{2} \right)^0 = 1000 \cdot 0 \cdot 1 = 0
\]
2. **Calculate \( f(1) \)**:
\[
f(1) = 1000 \cdot 1 \cdot \left( \frac{1}{2} \right)^1 = 1000 \cdot 1 \cdot \frac{1}{2} = 500
\]
3. **Calculate \( f(2) \)**:
\[
f(2) = 1000 \cdot 2 \cdot \left( \frac{1}{2} \right)^2 = 1000 \cdot 2 \cdot \frac{1}{4} = 500
\]
4. **Calculate \( f(3) \)**:
\[
f(3) = 1000 \cdot 3 \cdot \left( \frac{1}{2} \right)^3 = 1000 \cdot 3 \cdot \frac{1}{8} = 375
\]
Thus, the values are:
- \( f(0) = 0 \)
- \( f(1) = 500 \)
- \( f(2) = 500 \)
- \( f(3) = 375 \)
Now, we can compare the pattern results from the response options:
- **A**: 1000, 500, 250, 125
- **B**: 1000, 2000, 4000, 8000
- **C**: 1000, 999.5, 999, 998.5
- **D**: 1000, 1000.5, 1001, 1001.5
None of the provided options match the calculated values \( 0, 500, 500, 375 \).
If you want to check the conditions or the values provided or if it seems that the calculation intervals should be different, please clarify the function or the expected output.
1. **Calculate \( f(0) \)**:
\[
f(0) = 1000 \cdot 0 \cdot \left( \frac{1}{2} \right)^0 = 1000 \cdot 0 \cdot 1 = 0
\]
2. **Calculate \( f(1) \)**:
\[
f(1) = 1000 \cdot 1 \cdot \left( \frac{1}{2} \right)^1 = 1000 \cdot 1 \cdot \frac{1}{2} = 500
\]
3. **Calculate \( f(2) \)**:
\[
f(2) = 1000 \cdot 2 \cdot \left( \frac{1}{2} \right)^2 = 1000 \cdot 2 \cdot \frac{1}{4} = 500
\]
4. **Calculate \( f(3) \)**:
\[
f(3) = 1000 \cdot 3 \cdot \left( \frac{1}{2} \right)^3 = 1000 \cdot 3 \cdot \frac{1}{8} = 375
\]
Thus, the values are:
- \( f(0) = 0 \)
- \( f(1) = 500 \)
- \( f(2) = 500 \)
- \( f(3) = 375 \)
Now, we can compare the pattern results from the response options:
- **A**: 1000, 500, 250, 125
- **B**: 1000, 2000, 4000, 8000
- **C**: 1000, 999.5, 999, 998.5
- **D**: 1000, 1000.5, 1001, 1001.5
None of the provided options match the calculated values \( 0, 500, 500, 375 \).
If you want to check the conditions or the values provided or if it seems that the calculation intervals should be different, please clarify the function or the expected output.