Questions: The function g is defined by the following rule. g(x) = (1/10)^x Find g(x) for each x-value in the table. x g(x) -3 999.999999 -2 99.99999 -1 10 0 1.0 1 0.1

The function g is defined by the following rule. g(x) = (1/10)^x Find g(x) for each x-value in the table. x g(x) -3 999.999999 -2 99.99999 -1 10 0 1.0 1 0.1
Transcript text: The function $g$ is defined by the following rule. \[ g(x)=\left(\frac{1}{10}\right)^{x} \] Find $g(x)$ for each $x$-value in the table. \begin{tabular}{|c|c|} \hline$x$ & $g(x)$ \\ \hline-3 & 999.999999 \\ \hline-2 & 99.99999 \\ \hline-1 & 10 \\ \hline 0 & 1.0 \\ \hline 1 & $0.1]^{n}$ \\ \hline \end{tabular}
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Solution

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Solution Steps

To find \( g(x) \) for each \( x \)-value in the table, we need to evaluate the function \( g(x) = \left(\frac{1}{10}\right)^{x} \) for each given \( x \). This involves computing the power of \( \frac{1}{10} \) raised to the \( x \)-th power for each \( x \) value in the table.

Step 1: Evaluate \( g(-3) \)

To find \( g(-3) \), we calculate: \[ g(-3) = \left(\frac{1}{10}\right)^{-3} = 10^3 = 1000 \] However, the output shows \( g(-3) \approx 999.9999999999999 \), which we can round to \( 1000 \).

Step 2: Evaluate \( g(-2) \)

Next, we calculate \( g(-2) \): \[ g(-2) = \left(\frac{1}{10}\right)^{-2} = 10^2 = 100 \] The output shows \( g(-2) \approx 99.99999999999999 \), which we can round to \( 100 \).

Step 3: Evaluate \( g(-1) \)

Now, we find \( g(-1) \): \[ g(-1) = \left(\frac{1}{10}\right)^{-1} = 10^1 = 10 \] The output confirms \( g(-1) = 10.0 \).

Step 4: Evaluate \( g(0) \)

Next, we calculate \( g(0) \): \[ g(0) = \left(\frac{1}{10}\right)^{0} = 1 \] The output confirms \( g(0) = 1.0 \).

Step 5: Evaluate \( g(1) \)

Finally, we find \( g(1) \): \[ g(1) = \left(\frac{1}{10}\right)^{1} = 0.1 \] The output confirms \( g(1) = 0.1 \).

Final Answer

The values of \( g(x) \) for the given \( x \)-values are:

  • \( g(-3) \approx 1000 \)
  • \( g(-2) \approx 100 \)
  • \( g(-1) = 10 \)
  • \( g(0) = 1 \)
  • \( g(1) = 0.1 \)

Thus, the final boxed answers are: \[ \boxed{g(-3) \approx 1000} \] \[ \boxed{g(-2) \approx 100} \] \[ \boxed{g(-1) = 10} \] \[ \boxed{g(0) = 1} \] \[ \boxed{g(1) = 0.1} \]

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