To determine what kind of function is represented by the given values, we can observe the pattern in the outputs \( f(x) \) corresponding to the inputs \( x \).
The table shows the following values:
- \( f(-3) = -256 \)
- \( f(-2) = -128 \)
- \( f(-1) = -64 \)
- \( f(0) = -32 \)
- \( f(1) = -16 \)
We can see that the values of \( f(x) \) are doubling as we move from \( x = -3 \) to \( x = 1 \).
More specifically, let’s look at the absolute values of \( f(x) \):
- From \( f(-3) = -256 \) to \( f(-2) = -128 \): \( -256 \div -128 = 2 \)
- From \( f(-2) = -128 \) to \( f(-1) = -64 \): \( -128 \div -64 = 2 \)
- From \( f(-1) = -64 \) to \( f(0) = -32 \): \( -64 \div -32 = 2 \)
- From \( f(0) = -32 \) to \( f(1) = -16 \): \( -32 \div -16 = 2 \)
This indicates that the function has a geometric progression where each step corresponds to \( f(x) = -256 \times (1/2)^{x + 3} \).
Thus, the function \( f(x) \) is an exponential function.
Final Answer:
This function is exponential because the outputs are decreasing by a consistent factor as the inputs increase, which is characteristic of exponential decay.