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Understanding how to compute function values is essential when dealing with exponential functions. The given function in our problem is \(r(x) = 2^{x} - 1\), and computing its values involves substituting the \( x \) values into the function and simplifying. For example, with \( x = -1 \), the function becomes \(r(-1) = 2^{-1} - 1\), which simplifies to \( \frac{1}{2} - 1 \), giving us \( -\frac{1}{2} \).

This process is repeated for each \( x \) value in the table. It's important to remember to apply the exponent first as per the order of operations—parentheses, exponents, multiplication and division, and then addition and subtraction (PEMDAS). This process yields the function's behavior or output, which is crucial for understanding how the function behaves across different \( x \) values. With exponential functions, as \( x \) increases, the function's output grows exponentially, unless modified by subtraction or other operations included in the function, like the '-1' in our formula.

Creating a table of values is a helpful way to visualize the relationship between \( x \) and \( r(x) \) in a function. For the function \( r(x) = 2^{x} - 1 \), we plot \( x \) values (inputs) and the corresponding \( r(x) \) values (outputs). This visualization helps us see patterns and can guide our understanding of how the function behaves.

In our example, negative \( x \) values resulted in negative \( r(x) \) outcomes due to the '-1' in the formula, while the positive \( x \) values naturally led to positive results after the subtraction. Once filled in, the table serves not just as a completed homework task, but also as a learning tool to predict and understand outputs for inputs that may not be explicitly calculated yet. Understanding how to read and interpret these tables is crucial for students learning about exponential functions and for troubleshooting any potential errors in their calculations.

Once the technology formula is input into the software, students can effortlessly plot an entire series of points. This not only confirms the calculated table values but also helps students make connections between abstract algebraic concepts and tangible visual representations. In our modern learning environments, integrating technology enhances comprehension and engages students in interactive and dynamic ways.