Question

In Exercises 47-50, (a) use a graphing utility to graph the function, (b) use the drawing feature of the graphing utility to draw the inverse of the function, and (c) determine whether the graph of the inverse relation is an inverse function. Explain your reasoning. $$ f(x)=x^{3}+x+4 $$

Short answer

The function \(f(x) = x^{3}+x+4\) is graphed first, followed by its inverse by reflecting it about the line \(y = x\). Its inverse will also be a function, as the graph passes the horizontal line test, i.e., no horizontal line intersects the graph at more than one point.

Step-by-step solution

Graphing the function

Firstly, using a graphing utility such as a graphing calculator or online graphing tool, the function \(f(x) = x^{3}+x+4\) needs to be plotted. Since it's a cubic function, the graph won't be a straight line.

Sketching the inverse

After plotting the function, the inverse can be plotted by reflecting the original graph about the line \(y = x\). This task can be performed using the drawing feature on a graphing calculator or graphing utility.

Determining if the inverse is a function

To identify if the graph of the inverse is a function, the horizontal line test can be applied. If any horizontal line crosses the inverse graph more than once, the inverse is not a function. Otherwise, it is a function.

Key concepts

Cubic Functions

Cubic functions are a type of polynomial function with the general form of f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a ≠ 0. They are characterized by their distinctive S-shaped curves and have the property of being symmetric with respect to their point of inflection, which is the point where the function changes concavity.

In the given exercise, the function f(x) = x^3 + x + 4 is a cubic function. When graphed, it shows the typical behavior of cubic polynomials, featuring one or more turns and a single point of inflection. It's important to recognize these turns and the point of inflection, especially when trying to sketch the inverse function since these features will be reflected across the line y = x.

Understanding cubic functions is crucial because they can model real-world phenomena, such as the relationship between variables in physics or economics, and provide insights into rate changes and trends.

Graphing Calculators

Graphing calculators are indispensable tools in mathematics, especially when dealing with complex functions like the cubic function discussed earlier. They allow for efficient and accurate plotting of equations that would be time-consuming and difficult to draw by hand.

To use a graphing calculator for the exercise at hand, you'd enter the function f(x) = x^3 + x + 4 into the calculator, which would then display the corresponding graph. These calculators not only plot the function but also offer features to zoom in and out, adjust the viewing window, and trace along the curve to find points of interest, such as the y-intercept, x-intercepts, and the point of inflection.

Moreover, advanced calculators may have functionalities to draw inverses directly. By utilizing this feature, you can easily visualize and understand the relationship between a function and its inverse.

Horizontal Line Test

The horizontal line test is a method used to determine if a function has an inverse that is also a function. This test is performed by drawing horizontal lines through the graph of a function.

If any horizontal line intersects the graph in more than one place, the function does not pass the horizontal line test, indicating that it is not one-to-one, and therefore it does not have an inverse that is also a function. On the other hand, if every horizontal line intersects the graph at most once, the function has an inverse that is a function.

Applying this to our cubic function example f(x) = x^3 + x + 4, if we graph its inverse and use the horizontal line test, we can determine whether the inverse is a function. The cubic function in question is unique in that it is one-to-one, and therefore its inverse will pass the horizontal line test, unlike most cubic functions which typically fail this test due to their multiple x-intercepts.