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Graphing functions can be a powerful tool to understand how they behave visually. By plotting points on a coordinate plane, we can see the shape and direction of the function. For example, if we are graphing a parabolic function like the one given in this exercise, we plot points for various values of the independent variable (usually x) and connect them to form the curve. This helps in interpreting the behavior and key features of the function. When graphing an inverse function, it's essential to plot points and connect them similarly. The inverse function essentially 'reverses' the operation of the original function. For the function and its inverse to be accurately represented, they should be graphically plotted on the same axes for direct comparison.

A parabola is a U-shaped curve that is symmetric around its axis of symmetry. The standard form of a parabolic function is either \( f(x) = ax^2 + bx + c \) or \( f(x) = a(x-h)^2 + k \) depending on how the expression is written. The function \( f(x) = x^2 - 1 \) considered on the interval \( x \leq 0 \) is a parabolic function that opens upwards with its vertex at the point (0, -1). Since the domain is restricted to \( x \leq 0 \), only the left half of the parabola is considered. When plotting a parabolic function, it is important to identify key points, such as the vertex, and plot them along with additional points along the curve. This will visually show the curvature and direction of the parabola.

Symmetry is a crucial concept in graphing functions and their inverses. A function and its inverse are symmetric with respect to the identity line y = x. This means if you fold a graph along this line, the function and its inverse would line up perfectly. In this exercise, after graphing the function and its inverse, drawing the line y = x helps verify their symmetry. For instance, the function f(x) = x^2 - 1 and its inverse f^(-1)(x) = -sqrt(x + 1) are mirror images across the line y = x. Understanding this symmetry helps to confirm that you've graphed the inverse correctly.