Question

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

$x$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$-13$	$-52$	$-208$	$-832$	$-3328$

answer attempt 1 out of 2
this function is \\(\quad\\) because

Explanation:

Step1: Check first differences

Calculate $\Delta f(x) = f(x_{n+1}) - f(x_n)$:
$\Delta f(-1\to0) = -52 - (-13) = -39$
$\Delta f(0\to1) = -208 - (-52) = -156$
$\Delta f(1\to2) = -832 - (-208) = -624$
$\Delta f(2\to3) = -3328 - (-832) = -2496$
First differences are not constant.

Step2: Check common ratios

Calculate $r = \frac{f(x_{n+1})}{f(x_n)}$:
$r_{-1\to0} = \frac{-52}{-13} = 4$
$r_{0\to1} = \frac{-208}{-52} = 4$
$r_{1\to2} = \frac{-832}{-208} = 4$
$r_{2\to3} = \frac{-3328}{-832} = 4$
The ratio between consecutive $f(x)$ values is constant.

Answer:

This function is an exponential function because the ratio of consecutive output values (for equally spaced input values) is a constant 4, which is the defining characteristic of an exponential function.