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Evaluating a function involves calculating the value of a function's output for a given input. In the exercise, we are working with the function \(f(x) = 3^{-x}\). The goal is to find \(f(x)\) for each specified value of \(x\). This is done by substituting each \(x\) value into the function and solving. For instance, to evaluate \(f(x)\) when \(x\) is 2, substitute 2 into the equation: - \(f(2) = 3^{-2}\)- This simplifies to \(f(2) = \frac{1}{9}\).This substitution method allows you to compute the function's value systematically for each \(x\) in the table. Remember: replace \(x\) with each number in your list and perform the necessary arithmetic operations to find the corresponding \(f(x)\).

Negative exponents can be tricky, but they are quite simple once you understand the rule. A negative exponent, such as in \(3^{-x}\), indicates a reciprocal. This means \(3^{-x}\) can be rewritten as \(\frac{1}{3^x}\).Here is how you handle negative exponents:

• For \( 3^{-1} \), it becomes \( \frac{1}{3^1} = \frac{1}{3} \).
• For \( 3^{-2} \), it becomes \( \frac{1}{3^2} = \frac{1}{9} \).
• For \( 3^{-3} \), it turns into \( \frac{1}{3^3} = \frac{1}{27} \).

This transformation from a negative exponent to a fraction form allows you to evaluate exponential functions more easily, especially when graphing or calculating manually.

Graphing calculators are invaluable tools for students and educators alike. They handle complex calculations, plot graphs, and evaluate functions easily. With the function \(f(x) = 3^{-x}\), a graphing calculator can help visualize the function's behavior across a range of \(x\) values. To use a graphing calculator to find \(f(x)\) values:

• Input the function into the calculator.
• Use the table function to quickly generate \(f(x)\) values for your chosen \(x\).
• Plot the graph to see how \(f(x)\) changes as \(x\) varies.

Graphing calculators like the TI-84 or online tools such as Desmos make it easy to visualize and understand exponential functions. This tool is especially useful for checking manual calculations.

Completing a table for a function involves organizing inputs and their corresponding outputs neatly. In this exercise, our focus was the exponential function \(f(x) = 3^{-x}\), covering a set of specified \(x\) values. Here’s a step-by-step on completing the table:

• First compute \(f(x)\) for each \(x\) using substitution and solving, as described before.
• Next, write down each calculated value next to its corresponding \(x\) value.

The completed table should reflect:- \(x = -3\), \(f(x) = 27\)- \(x = -2\), \(f(x) = 9\)- \(x = -1\), \(f(x) = 3\)- \(x = 0\), \(f(x) = 1\)- \(x = 1\), \(f(x) = \frac{1}{3}\)- \(x = 2\), \(f(x) = \frac{1}{9}\)- \(x = 3\), \(f(x) = \frac{1}{27}\)Completing tables is a fundamental skill that reinforces understanding of functional relationships, especially in exponential functions.