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Understanding the domain and range of a function is crucial when graphing. For the function given by the equation \(y = x^{2/3} - 4\), the domain refers to all the possible values \(x\) can take. Since \(x^{2/3}\) is defined for every real number \(x\), the domain of this function is also all real numbers \((-\infty, \infty)\). This means that you can pick any real number for \(x\) and find a corresponding \(y\).
The range, on the other hand, is all the possible values \(y\) can take based on the function. While \(x^{2/3}\) always produces a non-negative result, subtracting 4 shifts the entire output downwards by 4 units. Thus, the range of \(f(x) = x^{2/3} - 4\) covers all real numbers as \(y\) can go from negative to positive infinity, following the value of \(x^{2/3} - 4\).

Finding key points on a graph helps sketch it accurately. These points are values of \(x\) for which we calculate \(y\) and they help indicate the shape and position of the curve.

• Starting with \(x = 0\), the equation becomes \(y = 0^{2/3} - 4 = -4\). This gives us the point \((0, -4)\).
• For \(x = 1\), substituting into the function gives \(y = 1^{2/3} - 4 = -3\), resulting in the point \((1, -3)\).
• Similarly, for \(x = -1\), the result is \(y = (-1)^{2/3} - 4 = -3\), providing the point \((-1, -3)\).

These key points are critical as they establish a basic framework from which the behavior of the function can be extrapolated.

An even function is one that shows symmetry around the y-axis. For the function \(f(x) = x^{2/3} - 4\), the term \(x^{2/3}\) is an even function, meaning it behaves the same way for positive and negative values of \(x\). This symmetry implies that the resulting graph does not change when reflected over the y-axis.
Even functions have a distinctive feature: \(f(x) = f(-x)\) for all values in their domain. In this exercise's context, when you substitute \(x\) and \(-x\) into the function, you end up with the same \(y\)-values, such as \(f(1) = -3\) and \(f(-1) = -3\). Recognizing this property helps understand the function's graphical behavior and predict values easily.

Symmetry in functions often makes them easier to understand and graph. The function \(y = x^{2/3} - 4\) shows symmetry about the y-axis because of its even function component. This symmetry reassures us that the graph on the left side of the y-axis mirrors that on the right side. Knowing this helps when plotting points, as every point on one side has a corresponding point directly opposite it on the other side.
This type of symmetry also means that the graph will reach its minimum or maximum at the y-axis itself. For this function, it reaches a minimum value at \(x = 0\), where \(y = -4\). As each side of the graph is a perfect mirror image, it simplifies the graphing process and verifies any calculations made for specific \(x\) values.