Question 1 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)=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 f of x is equal to 2 point 2 3 times 1 point 4 0 to the x th power ; f of negative 1 is equal to 1 point 5 9 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

1 answer

To determine the function that best models the data in the table and find \( f(-1) \), we can start by analyzing the pattern of the y-values corresponding to the x-values.

Here are the x and y values again for reference:

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

Notice the pattern:

  • \( 5 \) at \( x=0 \)
  • \( 15 = 5 \cdot 3 \) at \( x=1 \)
  • \( 45 = 15 \cdot 3 \) at \( x=2 \)
  • \( 135 = 45 \cdot 3 \) at \( x=3 \)
  • \( 405 = 135 \cdot 3 \) at \( x=4 \)

The pattern shows that y-values can be generated by multiplying the previous y-value by \( 3 \). Thus, it appears that the function is an exponential function.

The closest fitting function appears to be: \[ f(x) = 5 \cdot 3^x \]

Now, let's calculate \( f(-1) \): \[ f(-1) = 5 \cdot 3^{-1} = 5 \cdot \frac{1}{3} = \frac{5}{3} \]

Now, let's look at the responses given:

  1. \( f(x) = 92x - 63; f(-1) = -155 \)
  2. \( f(x) = 2.23 \cdot (1.40)^x; f(-1) = 1.59 \)
  3. \( f(x) = 5 \cdot 3^x; f(-1) = \frac{5}{3} \)
  4. \( f(x) = 41.43x^2 - 73.71x + 19.86; f(-1) = 135 \)

It seems the third option accurately reflects the data and calculated value. Therefore, the correct answer is:

Response: \( f(x) = 5 \cdot 3^x; f(-1) = \frac{5}{3} \)