Define the function $f: \mathbf{R} \rightarrow \mathbf{R}$ by $y=f(x)=x^{2}, x \in \mathbf{R}$. Complete the Table given below by using this definition. What is the domain and range of this function? Draw the graph of $f$. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline y=f(x)=x^{2} & & & & & & & & & \\ \hline \end{array}$$

Step 1: Understand the Function Definition

The problem defines a function $$f$$ from the set of real numbers $$\mathbf{R}$$ to the set of real numbers $$\mathbf{R}$$ , given by the equation $$y = f(x) = x^2$$ . This means for any input value $$x$$ , the output $$y$$ is the square of $$x$$ .

Step 2: Complete the Table

To complete the table, we substitute each given $$x$$ value into the function $$f(x) = x^2$$ and calculate the corresponding $$y$$ value. For example, if $$x = -4$$ , then $$y = (-4)^2 = 16$$ . If $$x = 3$$ , then $$y = (3)^2 = 9$$ .

Step 3: Determine Domain and Range

The domain of a function is the set of all possible input values ( $$x$$ ). Since we can square any real number, the domain of $$f(x) = x^2$$ is all real numbers, denoted by $$\mathbf{R}$$ . The range is the set of all possible output values ( $$y$$ ). Since the square of any real number is always non-negative, the range of $$f(x) = x^2$$ is all non-negative real numbers, which can be written as $$[0, \infty)$$ .

Step 4: Draw the Graph of the Function

Using the points from the completed table, we can plot them on a coordinate plane. Connecting these points with a smooth curve reveals that the graph of $$f(x) = x^2$$ is a parabola. This parabola opens upwards, is symmetric about the $$y$$ -axis, and has its lowest point, or vertex, at the origin $$(0{,}0)$$ .

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