2. Each of the following tables shows a collection of points $(x, f(x))$ from a function $f$. For each table, determine whether the $f$ is possibly linear, possibly exponential, or cannot be either.

If you decide that $f$ is exponential or linear, provide the rule for the function.
(a)
\begin{tabular}{|l|l|l|l|l|l|}
\hline$x$ & -1 & 0 & 1 & 2 & 3 \
\hline$f(x)$ & 5 & 2 & -1 & -4 & -7 \
\hline
\end{tabular}
(b)
\begin{tabular}{|l|l|l|l|l|l|}
\hline$x$ & -1 & 0 & 1 & 2 & 3 \
\hline$f(x)$ & 32 & 16 & 8 & 4 & 2 \
\hline
\end{tabular}
(c)
\begin{tabular}{|l|l|l|l|l|l|}
\hline$x$ & -1 & 0 & 1 & 2 & 3 \
\hline$f(x)$ & 2 & 4 & 7 & 11 & 16 \
\hline
\end{tabular}
[3 marks]

Question

2. Each of the following tables shows a collection of points $(x, f(x))$ from a function $f$. For each table, determine whether the $f$ is possibly linear, possibly exponential, or cannot be either.

If you decide that $f$ is exponential or linear, provide the rule for the function.
(a)
\begin{tabular}{|l|l|l|l|l|l|}
\hline$x$ & -1 & 0 & 1 & 2 & 3 \
\hline$f(x)$ & 5 & 2 & -1 & -4 & -7 \
\hline
\end{tabular}
(b)
\begin{tabular}{|l|l|l|l|l|l|}
\hline$x$ & -1 & 0 & 1 & 2 & 3 \
\hline$f(x)$ & 32 & 16 & 8 & 4 & 2 \
\hline
\end{tabular}
(c)
\begin{tabular}{|l|l|l|l|l|l|}
\hline$x$ & -1 & 0 & 1 & 2 & 3 \
\hline$f(x)$ & 2 & 4 & 7 & 11 & 16 \
\hline
\end{tabular}
[3 marks]

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ For table (a), we observe the values of $f(x)$: 5, 2, -1, -4, -7. The differences between consecutive values are: $2 - 5 = -3$, $-1 - 2 = -3$, $-4 - (-1) = -3$, and $-7 - (-4) = -3$. Since the first differences are constant, the function is possibly linear. ‖ ‖ step 2 ⋮ The linear function can be expressed in the form $f(x) = mx + b$. Using the points (0, 2) and the slope $m = -3$, we can find the equation. The equation becomes $f(x) = -3x + 2$. ‖ ‖ step 3 ⋮ For table (b), the values of $f(x)$ are: 32, 16, 8, 4, 2. The ratios of consecutive values are: $16/32 = 1/2$, $8/16 = 1/2$, $4/8 = 1/2$, and $2/4 = 1/2$. Since the ratios are constant, the function is possibly exponential. ‖ ‖ step 4 ⋮ The exponential function can be expressed in the form $f(x) = ab^x$. Using the point (0, 32) and the ratio $b = 1/2$, we find $f(x) = 32(1/2)^x$. ‖ ‖ step 5 ⋮ For table (c), the values of $f(x)$ are: 2, 4, 7, 11, 16. The differences between consecutive values are: $4 - 2 = 2$, $7 - 4 = 3$, $11 - 7 = 4$, and $16 - 11 = 5$. The differences are not constant, and the second differences are also not constant, indicating that the function is neither linear nor exponential. ‖ ∻Answer∻ ⚹ (a) Linear: $f(x) = -3x + 2$; (b) Exponential: $f(x) = 32(1/2)^x$; (c) Neither ⚹ ∻Key Concept∻ ⚹ Identifying function types based on data points ⚹ ∻Explanation∻ ⚹ By analyzing the differences and ratios of the function values, we can determine if the function is linear, exponential, or neither, and derive the corresponding equations when applicable. ⚹◊What is the general form of a linear function and how can you determine if a set of points fits this form?⍭ Generate me a similar question◊

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