Question

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

$x$	$-1$	$0$	$1$	$2$	$3$

answer attempt 1 out of 2

this function is because
.

Explanation:

Step 1: Check for common ratio (exponential) or common difference (linear)

First, let's check the ratios between consecutive \( f(x) \) values.

For \( x = -1 \) to \( x = 0 \): \( \frac{528}{352}=\frac{3}{2} = 1.5 \)

For \( x = 0 \) to \( x = 1 \): \( \frac{792}{528}=\frac{3}{2}=1.5 \)

For \( x = 1 \) to \( x = 2 \): \( \frac{1188}{792}=\frac{3}{2}=1.5 \)

For \( x = 2 \) to \( x = 3 \): \( \frac{1782}{1188}=\frac{3}{2}=1.5 \)

Now check the differences:

\( 528 - 352 = 176 \)

\( 792 - 528 = 264 \)

\( 1188 - 792 = 396 \)

\( 1782 - 1188 = 594 \)

The differences are not constant (\( 176, 264, 396, 594 \) have a common ratio of \( 1.5 \) as well, but the key is the ratios between \( f(x) \) values are constant).

Step 2: Determine the function type

Since the ratio of consecutive \( f(x) \) values is constant (\( 1.5 \) or \( \frac{3}{2} \)), the function is exponential. The reason is that in an exponential function, \( f(x)=a\cdot b^x \), and the ratio \( \frac{f(x + 1)}{f(x)}=b \) (constant), which matches our calculation here.

Answer:

This function is \(\boldsymbol{\text{exponential}}\) because the ratio of consecutive \( f(x) \) values is constant (each time \( f(x) \) is multiplied by \( \frac{3}{2} \) or \( 1.5 \) to get the next \( f(x) \) value).