Question

Use the table to answer the question.



x-Values y-Values
0 5
1 15
2 45
3 135
4 405
Select the function that best models the data and determine the value of the function at x=−1
.

(1 point)
Responses

f(x)=5⋅3x
; f(−1)=53
f of x is equal to 5 times 3 to the x th power; f of negative 1 is equal to 5 thirds

f(x)=92x−63
; f(−1)=−155
f of x is equal to 92 x minus 63; f of negative 1 is equal to negative 155

f(x)=41.43x2−73.71x+19.86
; f(−1)=135
f of x is equal to 41 point 4 3 x squared minus 73 point 7 1 x plus 19 point 8 6; f of negative 1 is equal to 135

f(x)=2.23⋅(1.40)x
; f(−1)=1.59

Answer 1

To determine which function best models the given data, we can analyze the y-values in relation to the x-values and look for a pattern.

Evaluating the given data:
- For \( x = 0 \), \( y = 5 \)
- For \( x = 1 \), \( y = 15 \)
- For \( x = 2 \), \( y = 45 \)
- For \( x = 3 \), \( y = 135 \)
- For \( x = 4 \), \( y = 405 \)

We can notice that each y-value seems to be increasing by factors that may suggest an exponential or polynomial function.

Let’s consider the function \( f(x) = 5 \cdot 3^x \):
- For \( x = 0 \), \( f(0) = 5 \cdot 3^0 = 5 \)
- For \( x = 1 \), \( f(1) = 5 \cdot 3^1 = 15 \)
- For \( x = 2 \), \( f(2) = 5 \cdot 3^2 = 45 \)
- For \( x = 3 \), \( f(3) = 5 \cdot 3^3 = 135 \)
- For \( x = 4 \), \( f(4) = 5 \cdot 3^4 = 405 \)

This function indeed matches all the given data points.

Now to determine the value of the function at \( x = -1 \):
\[
f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3}
\]

Thus, the correct response is:
**f(x) = 5 ⋅ 3^x; f(−1) = 5/3**.