Question

What is the rule that defines the function represented by the table?

x 0 1 2 3 4
y 40 10 52 58 532
A. 𝑓(𝑥)=14(40)𝑥
B. 𝑓(𝑥)=12(40)𝑥
C. 𝑓(𝑥)=40(14)𝑥
D. 𝑓(𝑥)=4(13)𝑥

Answer 1

To determine which of the given functions represents the relationship between \( x \) and \( y \) in the table, let’s calculate the values of \( f(x) \) for each of the given options and compare them to the values in the table.

Given data:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 40 \\
1 & 10 \\
2 & 52 \\
3 & 58 \\
4 & 532 \\
\hline
\end{array}
\]

Now, let's analyze each option.

### Option A: \( f(x) = 14(40)^x \)

- For \( x = 0 \): \( f(0) = 14(40)^0 = 14 \times 1 = 14 \) (not equal to 40)
- This option does not fit.

### Option B: \( f(x) = 12(40)^x \)

- For \( x = 0 \): \( f(0) = 12(40)^0 = 12 \times 1 = 12 \) (not equal to 40)
- This option does not fit.

### Option C: \( f(x) = 40(14)^x \)

- For \( x = 0 \): \( f(0) = 40(14)^0 = 40 \times 1 = 40 \) (equal to 40)
- For \( x = 1 \): \( f(1) = 40(14)^1 = 40 \times 14 = 560 \) (not equal to 10)
- This option does not fit.

### Option D: \( f(x) = 4(13)^x \)

- For \( x = 0 \): \( f(0) = 4(13)^0 = 4 \times 1 = 4 \) (not equal to 40)
- This option does not fit.

Given this analysis, none of the options fit the values in the table completely.

It’s clear that the functions provided are not the correct ones for the given values of \( x \) and \( y \). If there could be a typo or if the function form is different, please provide additional details for clarification.