Math  /  Algebra

QuestionDetermine whether the function given in the table is linear, exponential, or neither. If the function is linear, find a linear function that models the data; if it is exponential, find an exponential function that models the data. \begin{tabular}{|c|c|} \hline x\mathbf{x} & f(x)\mathbf{f}(\mathbf{x}) \\ \hline-1 & 1 \\ \hline 0 & 6 \\ \hline 1 & 11 \\ \hline 2 & 16 \\ \hline 3 & 21 \\ \hline \end{tabular}
Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The function is exponential. An exponential function that models the data is f(x)=f(x)= \square \square. (Simplify your answer.) B. The function is linear. A linear function that models the data is f(x)=5x+4f(x)=5 x+4. \square (Simplify your answer.).

Studdy Solution

STEP 1

1. The function is either linear, exponential, or neither.
2. A linear function has a constant rate of change.
3. An exponential function has a constant ratio between consecutive outputs.

STEP 2

1. Determine if the function is linear by checking for a constant rate of change.
2. Determine if the function is exponential by checking for a constant ratio.
3. Conclude whether the function is linear, exponential, or neither.
4. If linear, find the linear function that models the data.
5. If exponential, find the exponential function that models the data.

STEP 3

Calculate the differences between consecutive f(x) f(x) values to check for a constant rate of change.
\begin{align*} f(0) - f(-1) &= 6 - 1 = 5, \\ f(1) - f(0) &= 11 - 6 = 5, \\ f(2) - f(1) &= 16 - 11 = 5, \\ f(3) - f(2) &= 21 - 16 = 5. \end{align*}
The differences are constant.

STEP 4

Since we found a constant difference, check for a constant ratio to confirm if it is not exponential.
Calculate the ratios between consecutive f(x) f(x) values:
\begin{align*} \frac{f(0)}{f(-1)} &= \frac{6}{1} = 6, \\ \frac{f(1)}{f(0)} &= \frac{11}{6}, \\ \frac{f(2)}{f(1)} &= \frac{16}{11}, \\ \frac{f(3)}{f(2)} &= \frac{21}{16}. \end{align*}
The ratios are not constant.

STEP 5

Since the differences are constant and the ratios are not, the function is linear.

STEP 6

To find the linear function, use the point-slope form of a line. We know the slope m=5 m = 5 and a point on the line, say (0,6) (0, 6) .
The linear function is:
f(x)=5x+6f(x) = 5x + 6
The function is linear. A linear function that models the data is:
f(x)=5x+6 f(x) = 5x + 6

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