To determine which function best represents the given values in the table, we can look at the provided data and analyze it.
The table you've provided shows:
\[ \begin{align*} x & : 0 & 1 & 2 & 3 & 4 \ f(x) & : 2 & 8 & 32 & 128 & 512 \ \end{align*} \]
Now let's evaluate the suggested functions:
-
\( f(x) = 2 \cdot 4^x \):
- For \( x = 0 \): \( f(0) = 2 \cdot 4^0 = 2 \)
- For \( x = 1 \): \( f(1) = 2 \cdot 4^1 = 8 \)
- For \( x = 2 \): \( f(2) = 2 \cdot 4^2 = 32 \)
- For \( x = 3 \): \( f(3) = 2 \cdot 4^3 = 128 \)
- For \( x = 4 \): \( f(4) = 2 \cdot 4^4 = 512 \)
This function matches all the values in the table.
-
\( f(x) = x + 2 \):
- For this function:
- \( f(0) = 0 + 2 = 2 \)
- \( f(1) = 1 + 2 = 3 \)
- \( f(2) = 2 + 2 = 4 \)
- \( f(3) = 3 + 2 = 5 \)
- \( f(4) = 4 + 2 = 6 \)
This does not match the values in the table.
- For this function:
-
\( f(x) = 6x + 2 \):
- For this function:
- \( f(0) = 6 \cdot 0 + 2 = 2 \)
- \( f(1) = 6 \cdot 1 + 2 = 8 \)
- \( f(2) = 6 \cdot 2 + 2 = 14 \)
- \( f(3) = 6 \cdot 3 + 2 = 20 \)
- \( f(4) = 6 \cdot 4 + 2 = 26 \)
This does not match the values in the table as well.
- For this function:
-
\( f(x) = x^5 \):
- For this function:
- \( f(0) = 0^5 = 0 \)
- \( f(1) = 1^5 = 1 \)
- \( f(2) = 2^5 = 32 \)
- \( f(3) = 3^5 = 243 \)
- \( f(4) = 4^5 = 1024 \)
This also does not match the values in the table.
- For this function:
Conclusion:
The only function that accurately represents the values given in the table is:
\[ f(x) = 2 \cdot 4^x \]