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The domain and range are fundamental attributes of a function that provide crucial information about the set of inputs and possible outputs. When considering the function \( f(x) = 4 - x^{2/3} \), the domain refers to all possible values of \( x \) for which the function is defined.

For \( f(x) \), any real number can be substituted for \( x \) because the expression \( x^{2/3} \) is defined for every real number. Hence, the domain of this function is all real numbers, written as \((-\infty, \infty)\).

The range of the function, on the other hand, refers to all possible output values. By graphing this function, you observe that as \( x \) approaches very large positive or negative values, the function heads towards negative infinity. The highest point on the graph occurs when \( x = 0 \), giving \( f(0) = 4 \). Therefore, the range is \((-\infty, 4]\), indicating that the function outputs can go as low as negative infinity and as high as 4.

Identifying intervals where a function increases or decreases is vital for understanding its behavior. These intervals represent the x-values over which the function either climbs upwards or dips downwards.

For the function \( f(x) = 4 - x^{2/3} \), it's helpful to observe the graph to identify these intervals. You notice that when \( x < 0 \), the graph shows an upward trend for \( f(x) \), which means the function is increasing. This occurs due to the negative exponent effect, which results in an increasing trend. Thus, the interval of increase is \((-\infty, 0)\).

Conversely, when \( x > 0 \), the graph declines, indicating that \( f(x) \) is decreasing. This downward movement happens because the term \(-x^{2/3}\) adds a subtractive effect as \( x \) gets larger. Hence, the interval of decrease is \((0, \infty)\). Understanding these intervals enables you to sketch or predict the function's behavior over various domains.

Graphing calculators are powerful tools that help visualize mathematical functions and enhance understanding of their properties. To graph the function \( f(x) = 4 - x^{2/3} \), it's important to input the function correctly into the calculator and set an appropriate window to capture its behavior.

Begin by entering the function precisely as it is, ensuring that any operations follow the order of mathematics properly, especially with fractions and exponents. Adjust the viewing window to inspect how the function behaves for large positive and negative \( x \) values. This helps capture the full scope of the graph, from the peaks to the troughs.

• Set the y-axis to range from slightly below the lowest anticipated function value to slightly above the highest
• Set the x-axis to cover enough negative and positive values to observe the full trend

Using a graphing calculator can simplify the process of understanding complex functions by visually displaying properties like asymptotes, intercepts, and changes in direction.