Problem 50 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=1-x^{3}\)

Step-by-Step Solution

Answer

The inverse of \( f(x) = 1-x^3 \) is \( f^{-1}(x) = \sqrt[3]{1-x} \).

Step 1: Understand the Inverse Function

The inverse function of a function \( f \) is a function \( f^{-1} \) such that \( f(f^{-1}(x)) = x \) for every \( x \) in the domain of \( f^{-1} \). In simple terms, the function reverses the effect of \( f \). For our function, \( f(x) = 1 - x^3 \), we need to find \( y = f^{-1}(x) \) such that when we substitute \( y \) into \( f \), we get \( x \).

Step 2: Set Up the Equation

Start with the equation for \( y = f^{-1}(x) \). Substitute \( y \) into the original function: \( x = 1 - y^3 \). This equation expresses \( y \) as a function of \( x \), which we will solve to find our inverse function.

Step 3: Solve for the Inverse

Rearrange the equation \( x = 1 - y^3 \) to solve for \( y \). First, isolate the term \( y^3 \) by subtracting 1 from both sides: \( y^3 = 1 - x \). Then, take the cube root of both sides to solve for \( y \): \( y = \sqrt[3]{1 - x} \).

Step 4: Verify the Inverse Function

To verify our solution, check that \( f(f^{-1}(x)) = x \). Substitute \( y = \sqrt[3]{1-x} \) into the original function: \( f(y) = 1 - (\sqrt[3]{1-x})^3 = 1 - (1-x) = x \). Therefore, \( f(f^{-1}(x)) = x \), confirming that our inverse function is correct.

Key Concepts

Functions

Functions are fundamental in mathematics as they define a unique relationship between two sets – often referred to as the domain and the range.

• A function takes an input from the domain and maps it to an output in the range.
• Traditionally written as \( f(x) \), here \( f \) denotes the function and \( x \) represents the input variable.
• The equation describes how to calculate the output from a given input.

For example, if we have a function \( f(x) = 1 - x^3 \), each input value of \( x \) will result in a different output. The function describes a rule for how \( x \) is transformed, highlighting the power of functions in modeling relationships in mathematics, science, and engineering. Understanding the nature of functions helps decompose complex problems into manageable parts.

Cube Roots

The cube root is an inverse operation of cubing a number, similar to how the square root is the inverse of squaring. It helps us "undo" the process of raising a number to the power of three.

• The cube root of a number \( a \) is a value \( b \) such that \( b^3 = a \).
• It is often denoted by \( \sqrt[3]{a} \).
• Cube roots can be either real or complex, but here we focus on real cube roots.

When dealing with the function \( f(x) = 1 - x^3 \) and wanting to find its inverse, recognizing that we will be working with cube roots is key. Solving \( x = 1 - y^3 \) led us to the inverse \( y = \sqrt[3]{1-x} \), making cube roots essential for moving between the function and its inverse. Cube roots are handy tools in unraveling equations where the variable is raised to the third power.

Verification of Inverse Functions

Verification of inverse functions is a crucial step in determining the correctness of your solution. The process involves checking if applying the original function and its calculated inverse results in no change to the input variable.

• For any function \( f \), its inverse \( f^{-1} \) should satisfy the condition: \( f(f^{-1}(x)) = x \).
• This means if you put the output from the inverse function back into the original function, you should get your starting value \( x \).
• Verification reassures you that both the function and its inverse correctly undo each other's operations.

Using the example, we found \( f^{-1}(x) = \sqrt[3]{1-x} \). Substituting this back into \( f(x) = 1 - x^3 \) indeed results in \( x \), confirming the inverse function is accurate. Verifying this not only provides confidence in our solution but also solidifies understanding of how the inverse functions operate.