Question

which of the exponential growth functions has a multiplicative rate of change of 4?

$x$	$f(x)$
$0$	$1$
$1$	$4$
$2$	$16$

$x$	$f(x)$
$0$	$1$
$1$	$2$
$2$	$4$

$x$	$f(x)$
$0$	$1$
$1$	$8$
$2$	$64$

Explanation:

Step1: Recall multiplicative rate of change

The multiplicative rate of change between consecutive \( x \)-values (with a difference of \( 1 \)) is \( \frac{f(x + 1)}{f(x)} \). We check each table.

Step2: Check first table

For \( x = -1 \) to \( x = 0 \): \( \frac{f(0)}{f(-1)}=\frac{1}{\frac{1}{4}} = 4 \).
For \( x = 0 \) to \( x = 1 \): \( \frac{f(1)}{f(0)}=\frac{4}{1}=4 \).
For \( x = 1 \) to \( x = 2 \): \( \frac{f(2)}{f(1)}=\frac{16}{4}=4 \).

Step3: Check second table

For \( x = -1 \) to \( x = 0 \): \( \frac{1}{\frac{1}{2}} = 2 \).
For \( x = 0 \) to \( x = 1 \): \( \frac{2}{1}=2 \).
For \( x = 1 \) to \( x = 2 \): \( \frac{4}{2}=2 \). Rate is \( 2 \), not \( 4 \).

Step4: Check third table

For \( x = -1 \) to \( x = 0 \): \( \frac{1}{\frac{1}{8}} = 8 \).
For \( x = 0 \) to \( x = 1 \): \( \frac{8}{1}=8 \).
For \( x = 1 \) to \( x = 2 \): \( \frac{64}{8}=8 \). Rate is \( 8 \), not \( 4 \).

Answer:

The first table (with \( x \) values \(-1, 0, 1, 2\) and \( f(x) \) values \(\frac{1}{4}, 1, 4, 16\))