Problem 13 - Precalculus with Limits

Question

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). \(f(x) = \srqt[3]{x}\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = \sqrt[3]{x}\) is \(f^{-1}(x) = x^3\). After verification, it's established that both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) hold.

Step 1: Find the Inverse Function

The given function is \(f(x) = \sqrt[3]{x}\). The first step for finding the inverse function is to replace \(f(x)\) with \(y\), which gives us \(y = \sqrt[3]{x}\). Then we interchange the role of \(x\) and \(y\), resulting in \(x = \sqrt[3]{y}\). Now, we solve this equation for \(y\). To do this, we cube both sides of the equation, getting \(x^3 = y\) which is the inverse function, denoted as \(f^{-1}(x) = x^3\)

Step 2: Verify \(f(f^{-1}(x)) = x\)

The expression \(f(f^{-1}(x))\) means applying the function \(f\) to its inverse \(f^{-1}\). This effectively means replacing \(x\) in \(f(x)\) with \(f^{-1}(x)\). When we do this for our functions, we get \(f(f^{-1}(x)) = \sqrt[3]{(x^3)}\). The cube root of \(x^3\) is just \(x\), so we've verified that \(f(f^{-1}(x))\) does indeed equal \(x\).

Step 3: Verify \(f^{-1}(f(x)) = x\)

The expression \(f^{-1}(f(x))\) means applying the inverse function \(f^{-1}\) to the function \(f\). For our functions, this means replacing \(x\) in our inverse function with \(f(x)\), thus giving us \(f^{-1}(f(x)) = (\sqrt[3]{x})^3\). The cube of the cube root of \(x\) is also simply \(x\), completing the verification that \(f^{-1}(f(x)) = x\)

Key Concepts

Cube Root Function

The cube root function is fundamental in mathematics and is often symbolized as \(f(x) = \sqrt[3]{x}\). It involves finding a number which, when multiplied by itself twice more, gives the original number. Cube roots work similarly to square roots, except they involve three multiplications. The cube root symbol, \(\sqrt[3]{x}\), tells us to find such a number for \(x\).

• A cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
• Cube root functions are important in fields that involve geometry and algebra.
• These functions are also real-valued for all real numbers, meaning they can take both positive and negative numbers.

The graph of a cube root function is unique. It resembles an elongated S-shape, extending from the bottom left to the top right of the graph. This reflects the nature of cube roots to produce both positive and negative outputs.

Function Verification

Function verification is a crucial part of understanding inverse functions. When you find an inverse function, you should confirm that it is correct through verification. This involves testing if the composite of the function and its inverse returns the original input value. This process guarantees the reliability of the inverse function you found.

• For \(f(x)\) and \(f^{-1}(x)\) to be inverses, both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) need to hold true.
• This means if you start with a number, apply the function, and then use the inverse function, you should get back to your original number.

Verification steps help affirm that your calculations and understanding are correct. This type of analysis is an essential skill for students to control math problems fundamentally.

Solving Equations

Solving equations is at the heart of mathematics, often involving finding the value of a variable that makes an equation true. Inverse functions are a powerful tool in these operations. To solve equations that involve cube roots, you need to manipulate the equation so that the variable of interest is isolated.

• Start by transforming the equation to make the variable alone on one side of the equation.
• If your equation involves a cube root, cubing both sides will eliminate it, thus simplifying the equation.
• Once the variable is isolated, solving the equation becomes straightforward.

Cubing both sides of an equation is an effective method when dealing with cube roots. This process requires caution to ensure no solutions are lost through mathematical operations or assumptions.