Question

Find the inverse of \(f .\) Then sketch the graphs of \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=x^{3}+1 $$

Short answer

The inverse function of \(f(x) = x^3 + 1\) is \(f^{-1}(x) = \sqrt[3]{x - 1}\). To sketch both graphs, first sketch the graph of \(f(x)\) which is a cubic curve passing through the point (0, 1) with a horizontal tangent. Next, sketch the graph of \(f^{-1}(x)\) which is a reflection of the original curve across the line y = x, passing through the point (1, 0) with a vertical tangent.

Step-by-step solution

Find the inverse function of f(x)

Since we are given f(x) = x^3 + 1, we first need to interchange x and y: y = x^3 + 1 x = y^3 + 1 Next, we solve for y to find the inverse function f^(-1)(x): x - 1 = y^3 y = \sqrt[3]{x - 1} Therefore, the inverse function is f^(-1)(x) = \sqrt[3]{x - 1}.

Sketch the graph of f(x) and the graph of its inverse function f^(-1)(x)

To sketch both graphs, keep in mind that the graph of the inverse function is just a reflection of the graph of the original function across the line y = x. 1. Sketch the graph of f(x) = x^3 + 1: A cubic function with the general form f(x) = x^3 + c will have a horizontal tangent at x = 0, passing through the point (0, c). In this case, the graph of f(x) will pass through the point (0, 1) with a horizontal tangent. 2. Sketch the line y = x: This will aid in visualizing the reflection of the original graph to its inverse. 3. Sketch the graph of f^(-1)(x) = \sqrt[3]{x - 1}: Since the graph of the inverse function is a reflection about the line y = x, the graph of f^(-1)(x) will pass through the point (1, 0) (which is the reflection of (0, 1)) with a vertical tangent. Now, both the original function f(x) = x^3 + 1 and its inverse function f^(-1)(x) = \sqrt[3]{x - 1} should be sketched on the same axes, with f(x) being a cubic curve passing through the point (0, 1) with a horizontal tangent, and f^(-1)(x) having a vertical tangent at the point (1, 0). The two curves should be reflections of each other across the line y = x.

Key concepts

Graphing Inverse Functions

Graphing inverse functions involves a specific procedure to visualize how two functions relate to one another. The notion is that every point on the graph of a function has a corresponding point on its inverse that is diagonally opposite with respect to the line y = x. To graph the inverse function, one typically starts by graphing the original function. Then, the line y = x is plotted, serving as a 'mirror'. Reflecting the graph of the original function across this line will result in the graph of the inverse function.

It’s important to note that not all functions have an inverse. For a function to have an inverse, it must be bijective, meaning it is both injective (one-to-one) and surjective (onto). When graphing, this concept visually ensures that each horizontal line intersects the function's graph at most once, which ensures there’s a unique inverse at that point.

The graphing process helps in understanding the symmetry between a function and its inverse, and it emphasizes the concept of function pairs 'undoing' each other’s actions.

Cubic Functions

Cubic functions are polynomial functions with a degree of three, and they have the general form f(x) = ax^{3} + bx^{2} + cx + d, where a, b, c, and d are constants, and a ≠ 0. The most basic cubic function is f(x) = x^{3} which displays specific characteristics such as having one point of inflection and symmetrical behavior regarding the origin.

When sketching a cubic function such as f(x) = x^{3} + 1, as given in our exercise, the '+1' indicates a vertical shift of the entire graph one unit up. Cubic graphs are recognized for their 'S' shaped curves and for having either one or three real roots (x-intercepts). Furthermore, the graph of a cubic function always has two turning points, or none, contributing to its distinct 'wavy' shape. Cubic functions can model various physical phenomena and are fundamental in polynomial equation studies.

Reflection Across the Line y = x

Reflection across the line y = x is a geometric transformation that implies 'flipping' a figure over this line. It’s pivotal when understanding inverse functions because the graph of an inverse function is a reflection of the original function's graph over this line. When reflecting a point (a, b) across the line y = x, the coordinates switch places to become (b, a).

In the context of functions, if (x, y) lies on the graph of the original function, then (y, x) will lie on the graph of its inverse. This transformation keeps the distances intact but changes the orientation of the graphed function. This concept is instrumental in visually verifying if two functions are indeed inverses of each other.

Solving for Inverse Functions

Solving for the inverse of a function requires a step-by-step process to transform the function into its inversely corresponding form. The procedure involves swapping the x and y variables and then solving the resultant equation for y. This algebraic manipulation unveils the formula for the inverse function.

In the provided exercise, starting with f(x) = x^{3} + 1, the variables are interchanged, leading us to x = y^{3} + 1 after swapping. Then, by isolating y, we arrive at the inverse function: f^{-1}(x) = \( \(sqrt[3]{x - 1}\) \). Careful algebraic work is essential to ensure accuracy, especially when dealing with more complex functions. Understanding this process is critical in calculus, as it lays the groundwork for more advanced concepts such as function composition and finding derivatives of inverse functions.