Question

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\sqrt[3]{x}-\sqrt[3]{x+1} $$

Short answer

To plot the graph of the function \(f(x) = \sqrt[3]{x} - \sqrt[3]{x+1}\), choose an appropriate viewing window, such as \([-10, 10]\) for both \(x\)-values and \(y\)-values. The graph shows the general shape of the function near the origin, with no specific symmetries or critical points. Keep in mind that the answer is not unique, and exploring different windows can reveal more information about the function's behavior.

Step-by-step solution

Identify the domain and range of the function

The domain of the function is the set of all real numbers, since we can take the cube root of any real number. The range of the function will also be all real numbers, as the result of the cube root operation is also a real number.

Identify any symmetries or key points of the function

To find any symmetry, we can try to determine if the function is even or odd. Although the individual cube root functions (\(\sqrt[3]{x}\) and \(\sqrt[3]{x+1}\)) are odd functions, their difference is not necessarily an odd function, and there is no simple symmetry. For key points, we can find the critical points of the function by taking the derivative and finding where it's equal to zero or doesn't exist. However, the function is smooth and there are no obvious critical points, so we'll skip this step in this case.

Choose an appropriate viewing window for the graph

To choose an appropriate viewing window, we can start by looking at the graph for a small range of values, for instance, around the origin. Let's try the interval \([-10, 10]\) for both \(x\)-values and \(y\)-values. If the graph does not provide enough information or show interesting behavior, we can adjust the viewing window accordingly.

Plot the graph of the function

Using the interval \([-10, 10]\) for both \(x\)-values and \(y\)-values, we can now plot the graph of the function \(f(x) = \sqrt[3]{x} - \sqrt[3]{x+1}\). We may end up with a graph that shows the general shape of the function near the origin, with no specific symmetries or critical points. Note: The answer is not unique, as different viewing windows may offer various insights into the function's behavior. The chosen window provides a reasonable view of the function near the origin, but exploring different windows can reveal further information about the function's behavior at different points.

Key concepts

Domain and Range of a Function

Understanding the domain and range of a function is crucial for graphing and analyzing its behavior.

Domain refers to the set of all possible input values (usually 'x' values) for which the function is defined. In our exercise, the function is defined as f(x) = \( sqrt[3]{x} \) - \( sqrt[3]{x+1} \). Since we can take the cube root of any real number, the domain of this function is all real numbers, or \(\mathbb{R}\).

Range, on the other hand, refers to the set of all possible outputs (the 'y' values) that the function can produce. In analyzing the given function, it is observed that by taking the cube root of a real number, we cover all real numbers as the output as well, making the range also all real numbers (\(\mathbb{R}\)).

Establishing the domain and range helps immensely in plotting and understanding the entirety of the function's graph, as it provides bounds within which the graph will exist.

Symmetry in Functions

Symmetry in functions is a visual characteristic that can simplify understanding a function's graph and predicting its behavior.

For a function to be considered symmetric, it usually must meet the criteria of being either even (symmetric about the y-axis) or odd (symmetric about the origin). To investigate this, one might replace \(x\) with \(-x\) and examine the resultant expression.

However, as noted in the step-by-step solution, the function f(x) = \( sqrt[3]{x} \) - \( sqrt[3]{x+1} \) does not exhibit clear symmetrical properties. Despite the individual cube root functions being odd, their subtraction leads to a function that is not odd, and no even symmetry is present either. The absence of symmetry means that we can expect the function's graph to have different properties and behavior on either side of the origin or the y-axis.

Derivative and Critical Points

The derivative of a function reveals a lot about its nature, particularly concerning its slopes and critical points, which are the points where the function's graph has a horizontal tangent line or the derivative does not exist.

To find these critical points, one would calculate the derivative of the function and set it equal to zero to solve for 'x' values that correspond to horizontal tangents, or identify where the derivative is undefined. Critical points are essential because they're often associated with the function's extreme points, such as local maxima and minima.

For the given function, the step-by-step solution suggests that there are no obvious critical points, and because it is a smooth function, we can infer that the graph does not have sharp turns or corners where the derivative would be undefined or zero. Although the exercise does not delve into calculating the derivative, this is a critical step in more complex situations when analyzing a function's graph and identifying potential extremes.