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)=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)=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)=2.23⋅(1.40)x
; f(−1)=1.59

Answer 1

To determine the function that best models the data and calculate the value of the function at \( x = -1 \), we can first examine the provided data:

| x-Values | y-Values |
|----------|----------|
| 0 | 5 |
| 1 | 15 |
| 2 | 45 |
| 3 | 135 |
| 4 | 405 |

From observing the values, we can see that:

- \( y(0) = 5 \)
- \( y(1) = 15 = 5 \cdot 3 \)
- \( y(2) = 45 = 5 \cdot 3^2 \)
- \( y(3) = 135 = 5 \cdot 3^3 \)
- \( y(4) = 405 = 5 \cdot 3^4 \)

This suggests that the function could be modeled by the equation \( f(x) = 5 \cdot 3^x \).

Now, let's confirm if this function fits and calculate \( f(-1) \):

\[ f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3} \]

So the model \( f(x) = 5 \cdot 3^x \) is appropriate, and at \( x = -1 \), the function evaluates to:

\[ f(-1) = \frac{5}{3} \]

Thus, the selected response:

**f(x) = 5⋅3^x; f(−1) = 5/3**