Questions: The function (f) is defined by the following rule. [ f(x)=left(frac16right)^x ] Find (f(x)) for each (x)-value in the table. (x) (f(x)) -3 (square) -2 (square) -1 (square) 0 (square) 1 (square)

$The function (f) is defined by the following rule. [ f(x)=left(frac16right)^x ] Find (f(x)) for each (x)-value in the table. (x) (f(x)) -3 (square) -2 (square) -1 (square) 0 (square) 1 (square)$

Transcript text: The function $f$ is defined by the following rule. \[ f(x)=\left(\frac{1}{6}\right)^{x} \] Find $f(x)$ for each $x$-value in the table. \begin{tabular}{|c|c|} \hline$x$ & $f(x)$ \\ \hline-3 & $\square$ \\ \hline-2 & $\square$ \\ \hline-1 & $\square$ \\ \hline 0 & $\square$ \\ \hline 1 & $\square$ \\ \hline \end{tabular}

Solution

Solution Steps

To find $ f(x) $ for each $ x $ value in the table, we need to evaluate the function $ f(x) = \left(\frac{1}{6}\right)^x $ for each given $ x $. This involves raising $ \frac{1}{6} $ to the power of each $ x $ value.

Step 1: Evaluate $ f(-3) $

To find $ f(-3) $, we calculate: \[ f(-3) = \left(\frac{1}{6}\right)^{-3} = 6^3 = 216 \]

Step 2: Evaluate $ f(-2) $

Next, we calculate $ f(-2) $: \[ f(-2) = \left(\frac{1}{6}\right)^{-2} = 6^2 = 36 \]

Step 3: Evaluate $ f(-1) $

Now, we find $ f(-1) $: \[ f(-1) = \left(\frac{1}{6}\right)^{-1} = 6^1 = 6 \]

Step 4: Evaluate $ f(0) $

For $ f(0) $: \[ f(0) = \left(\frac{1}{6}\right)^{0} = 1 \]

Step 5: Evaluate $ f(1) $

Finally, we calculate $ f(1) $: \[ f(1) = \left(\frac{1}{6}\right)^{1} = \frac{1}{6} \approx 0.1667 \]

Final Answer

The values of $ f(x) $ for the given $ x $ values are:

$ f(-3) = 216 $
$ f(-2) = 36 $
$ f(-1) = 6 $
$ f(0) = 1 $
$ f(1) \approx 0.1667 $

Thus, the final boxed answers are: \[ \boxed{f(-3) = 216}, \quad \boxed{f(-2) = 36}, \quad \boxed{f(-1) = 6}, \quad \boxed{f(0) = 1}, \quad \boxed{f(1) \approx 0.1667} \]

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