Answer the questions below to determine what kind of function is depicted in the table below.

Question

xx minus, 3−3 minus, 2−2 minus, 1−1 00 11
f, of, xf(x) minus, 256−256 minus, 128−128 minus, 64−64 minus, 32−32 minus, 16−16
Answer
Attempt 1 out of 2

This function is
because
.

Answer 1

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.