Define the real valued function $f: \mathbf{R}-\{0\} \rightarrow \mathbf{R}$ defined by $f(x)=\frac{1}{x}$, $x \in \mathbf{R}-\{0\}$. Complete the Table given below using this definition. What is the domain and range of this function? $$\begin{array}{|l|c|c|c|c|c|c|c|c|c|} \hline x & -2 & -1.5 & -1 & -0.5 & 0.25 & 0.5 & 1 & 1.5 & 2 \\ \hline y=\frac{1}{x} & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ \hline \end{array}$$

Step 1: Understand the Function Definition

The given function is $$f(x) = \frac{1}{x}$$ . This means that for any input value $$x$$ , the output f(x) is its reciprocal. The domain is specified as $$\mathbf{R}-\{0\}$$ , which means all real numbers except $$0$$ , because division by zero is undefined.

Step 2: Calculate Function Values for Negative x

We substitute each negative value of $$x$$ from the table into the function $$f(x) = \frac{1}{x}$$ to find the corresponding $$y$$ values. For example, when $$x = -2$$ , $$f(-2) = \frac{1}{-2} = -0.5$$ .

Step 3: Calculate Function Values for Positive x

Similarly, we substitute each positive value of $$x$$ from the table into the function $$f(x) = \frac{1}{x}$$ to find the corresponding $$y$$ values. For example, when $$x = 0.25$$ , $$f(0.25) = \frac{1}{0.25} = 4$$ .

Step 4: Complete the Table

By combining all the calculated values, we can complete the given table. The values are rounded to two decimal places where necessary.

Step 5: Determine Domain and Range

The domain of the function $$f(x) = \frac{1}{x}$$ is given as all real numbers except $$0$$ , because division by zero is undefined. The range of the function is also all real numbers except $$0$$ , because f(x) can never be $$0$$ (since $$1/x = 0$$ has no solution) and can take any other real value.

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