Question

Find all the inverses associated with the following functions and state their domains. $$f(x)=(x+1)^{3}$$

Short answer

Question: Find the inverse function associated with the given function $$f(x)=(x+1)^3$$ and determine its domain. Answer: The inverse function is $$f^{-1}(x) = \sqrt[3]{x} - 1$$, and its domain is $$(-\infty,\infty)$$.

Step-by-step solution

Replace the function with y

First, let's replace the given function $$f(x)$$ with $$y$$: $$y=(x+1)^3$$

Solve the equation for x in terms of y

Now, we will solve the equation for $$x$$ in terms of $$y$$ by taking the cube root of both sides: $$\sqrt[3]{y} = \sqrt[3]{(x+1)^3}$$ $$\sqrt[3]{y} = x+1$$ Next, we isolate $$x$$ by subtracting $$1$$ from both sides: $$x = \sqrt[3]{y} - 1$$

Replace y with the inverse notation

Now, we can replace $$y$$ with the inverse notation, $$f^{-1}(x)$$: $$f^{-1}(x) = \sqrt[3]{x} - 1$$

Determine the domain of the inverse function

The domain of the inverse function is the range of the original function. Since the function $$f(x) = (x+1)^3$$ is a cubic function, it has no restrictions, and its range is the set of all real numbers. Therefore, the domain of the inverse function is the set of all real numbers, which can be expressed as: $$(-\infty,\infty)$$ In conclusion, the inverse function associated with the given function is $$f^{-1}(x) = \sqrt[3]{x} - 1$$ with a domain of $$(-\infty,\infty)$$.

Key concepts

Understanding the Inverse Function Domain

The concept of an inverse function is essential in mathematics, representing a reversal of another function's action. To grasp an inverse function's domain, one must first understand that the domain refers to the set of all possible input values that the function can accept. Consequently, the domain of the inverse function specifically involves the range of the original function. In other words, the outputs of the initial function become the permissible inputs for its inverse.

Let's take, for example, the function given in the exercise:

f(x) = (x+1)^3.

By inverting this function, we arrive at its inverse, f^{-1}(x), which essentially tells us what input would have been necessary to get a particular output from the original function.

• If f(a) = b, then f^{-1}(b) = a.
• The domain of f^{-1} is thus equivalent to the range of f.

In the exercise, the function f(x) is a cubic polynomial, which is continuous and can take all real numbers as an output, making the range (-∞, ∞). Therefore, this is precisely the domain of the inverse function f^{-1}(x).

Solving Cubic Equations

Solving cubic equations—equations where the highest exponent of the variable is three—can be complex, but understanding the process is key to tackling many mathematical problems. These equations can often be solved by finding the cubic root of a number, much like finding the square root for quadratic equations.

To solve a cubic equation like y = (x+1)^3, one must isolate the variable:

• Introduce a new variable to simplify the expression: let y = (x+1)^3.
• Extract the cubic root of both sides to eliminate the exponent on one side, giving ∛y = x+1.
• Finally, isolate x by subtracting one from both sides, resulting in x = ∛y - 1.

A cubic root operation can also provide multiple roots, especially when complex numbers are considered, but in real number contexts, a single real root is often the primary goal.

Cubic Root Properties

The properties of cubic roots are an extension of those of square roots, but with a few distinct differences. Understanding these properties is critical when working with cubic equations and their inverses.

The cubic root of a number a, written as ∛a, is the number that, when cubed, equals a. This definition leads to the fundamental property that (∛a)^3 = a. Some additional properties include:

• The cubic root of a product is the product of the cubic roots: ∛(ab) = ∛a * ∛b.
• For any real number a, there is a unique real number ∛a because every real number has exactly one real cubic root.
• The cubic root function is an odd function, meaning it preserves the sign of the input: ∛(-a) = -∛a.

In our exercise, the cubic root property allows us to simplify (x+1)^3 to x+1 by applying the cubic root to both sides of the equation, thereby finding the inverse function effectively.