Question

In Exercises 35–46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=-x^3+3 $$

Short answer

Yes, the inverse of \(f\) is a function. The inverse function is given by \(f^{-1}(x) = -1\cdot(x - 3)^{1/3}\).

Step-by-step solution

Apply the horizontal line test

Draw a sketch of the function \(f(x) = -x^3 + 3\). If any horizontal line intersects the graph of the function at more than one point, we can say that the function does not pass the horizontal line test, and hence its inverse is not a function. The graph of \(f(x) = -x^3 + 3\) is a downward opening cubic, which does not intersect a horizontal line at more than one point no matter where the line is drawn. Therefore, the inverse of the function \(f\) is a function.

Find the inverse function

To find the inverse function of \(f(x) = -x^3 + 3\), first replace \(f(x)\) with \(y\) to get \(y = -x^3 + 3\). Then, swap the roles of y and x to get \(x = -y^3 + 3\). Solve this for y as follows: move \(y^3\) to the left and \(x\) to the right to get \(y^3 = -(x-3)\). Now, to isolate y, take the cube root of both sides to obtain \(y = \(-1\)*\((x-3)^{(1/3)}\)\), which is the inverse function.

Key concepts

Understanding the Horizontal Line Test

The horizontal line test is a visual way to determine whether a function has an inverse that is also a function. A single horizontal line drawn across the graph should only intersect the graph at one point. If it crosses more than once at any location, the function does not have an inverse that is a unique function, because it violates the basic definition of a function: every input must have exactly one output.

Applying this test to cubic functions, especially monotonically increasing or decreasing ones, like in our exercise, will show that they pass the horizontal line test. This implies that the inverses are indeed functions. This particular verification step is crucial because it ensures that each value in the range of the original function corresponds to one and only one value in the domain of the inverse function.

Function Inverses Explained Simply

The inverse of a function essentially reverses the roles of the input and output. To find the inverse function, denoted as f^{-1}(x), the steps typically involve replacing f(x) with y, swapping x and y, and then solving for y as the new function expression.

After identifying that the function passes the horizontal line test, as we have with the provided cubic function, we proceed with finding the inverse. It's important to solve the equation carefully, while ensuring to perform inverse operations properly. In the case of cubic functions, this involves taking the cube root after moving all terms involving y to one side and the rest to the opposite side. The resulting equation represents the inverse function, which can now be re-labeled as f^{-1}(x).

Cubic Functions and Their Characteristics

Cubic functions are polynomial functions with the highest degree of three. The standard form is f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a is non-zero. They are known for their distinct 'S' shaped curve and possess a single point of inflection, where the curvature changes.

In the context of our exercise, the given function f(x) = -x^3 + 3 is a simple downward opening cubic curve with a y-intercept at (0, 3). These functions are always one-to-one, meaning they will always pass the horizontal line test, making their inverses functions as well. Moreover, their behavior at infinity should be kept in mind - as x approaches infinity, a cubic function goes to positive or negative infinity, depending on the leading coefficient.