Question
Find the exponential function that is the best fit for f(x) defined by the table below.
x
1
2
3
4
5
6
f(x)
2
4
8
16
32
64
Question content area bottom
Part 1
f(x)equals
enter your response here
(Use integers or decimals for any numbers in the expression.)
x
1
2
3
4
5
6
f(x)
2
4
8
16
32
64
Question content area bottom
Part 1
f(x)equals
enter your response here
(Use integers or decimals for any numbers in the expression.)
Answers
GPT-4o mini
To find the exponential function that fits the data points given in the table, we need to recognize the pattern of the data:
- \( f(1) = 2 \)
- \( f(2) = 4 \)
- \( f(3) = 8 \)
- \( f(4) = 16 \)
- \( f(5) = 32 \)
- \( f(6) = 64 \)
We can see that \( f(x) \) appears to double with each increasing integer value of \( x \). This suggests that the function could be in the form:
\[
f(x) = a \cdot b^x
\]
To find \( a \) and \( b \), we can utilize any of the points.
Looking closely at the actual values of \( f(x) \):
- \( f(1) = 2 \)
- \( f(2) = 4 = 2^2 \)
- \( f(3) = 8 = 2^3 \)
- \( f(4) = 16 = 2^4 \)
- \( f(5) = 32 = 2^5 \)
- \( f(6) = 64 = 2^6 \)
From this, we can see that \( f(x) = 2^x \).
So, the best fit exponential function is:
\[
f(x) = 2^x
\]
You can now enter this expression for the response.
- \( f(1) = 2 \)
- \( f(2) = 4 \)
- \( f(3) = 8 \)
- \( f(4) = 16 \)
- \( f(5) = 32 \)
- \( f(6) = 64 \)
We can see that \( f(x) \) appears to double with each increasing integer value of \( x \). This suggests that the function could be in the form:
\[
f(x) = a \cdot b^x
\]
To find \( a \) and \( b \), we can utilize any of the points.
Looking closely at the actual values of \( f(x) \):
- \( f(1) = 2 \)
- \( f(2) = 4 = 2^2 \)
- \( f(3) = 8 = 2^3 \)
- \( f(4) = 16 = 2^4 \)
- \( f(5) = 32 = 2^5 \)
- \( f(6) = 64 = 2^6 \)
From this, we can see that \( f(x) = 2^x \).
So, the best fit exponential function is:
\[
f(x) = 2^x
\]
You can now enter this expression for the response.