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

Question

xx 11 22 33 44 55
f, of, xf(x) 2222 4444 8888 176176 352352
Answer
Attempt 1 out of 2

This function is
because...?
.

Answer 1

To determine the kind of function represented in the table, we will analyze the values provided:

Values of \( x \):

• \( 1, 2, 3, 4, 5 \)

Corresponding values of \( f(x) \):

• \( 2222, 4444, 8888, 176176, 352352 \)

Let's observe the pattern in \( f(x) \):

1. Analyzing the growth of \( f(x) \):

   • \( f(1) = 2222 \)
   • \( f(2) = 4444 \) (which is \( 2 \times 2222 \))
   • \( f(3) = 8888 \) (which is \( 4 \times 2222 \))
   • \( f(4) = 176176 \)
   • \( f(5) = 352352 \)

2. Checking Multiplicative Patterns:

   • The function does not seem linear, as the differences between the function values increase rapidly.
   • To see if it’s quadratic or exponential, we might look at ratios or further computations.

3. Notice that from \( f(2) \) to \( f(3) \): The value approximately doubles each time leading from \( 2222 \) to \( 4444 \). The value of \( f(4) \) and \( f(5) \) seems to be growing much faster.

4. Explore \( f(x) \) for patterns: Calculate \( f(x)/f(x-1) \):

   • \( f(2)/f(1) = 4444/2222 = 2 \)
   • \( f(3)/f(2) = 8888/4444 = 2 \)
   • \( f(4)/f(3) \) and \( f(5)/f(4) \) would need to be calculated further, but jumps here seem huge suggesting rapid growth.

5. Determine Function Type: Since the growth is increasing at an exponential rate with multiplicative jumps:

• \( f(x) \) appears closer to an exponential function rather than a polynomial one.

Given all the observations, the function is likely exponential.

Answer

This function is exponential because the values of \( f(x) \) show rapid growth that is multiplicative rather than additive, suggesting the function increases at a rate proportional to its current value.