Problem 24 - College Algebra Essentials

Question

The functions in Exercises \(11-28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\) $$f(x)=\sqrt[3]{x}$$

Step-by-Step Solution

Answer

The inverse function for \(f(x) = \sqrt[3]{x}\) is \(f^{-1}(x) = x^3\). The verifications \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) hold true for this function and its inverse, confirming the inverse function is correct.

Step 1: Finding the Inverse Function

The given function is \(f(x) = \sqrt[3]{x}\). To find the inverse function \(f^{-1}(x)\), we should replace \(f(x)\) with \(y\), so we get \(y = \sqrt[3]{x}\). Then we interchange the positions of \(y\) and \(x\) to get \(x = \sqrt[3]{y}\), which, when cubed, gives us \(x^3 = y\), where \(y\) is the new function in terms of \(x\). Therefore, the inverse function is \(f^{-1}(x) = x^3\).

Step 2: Verification Part 1

We need to verify that \(f(f^{-1}(x)) = x\). For that, we take \(f^{-1}(x) = x^3\) and substitute it in the place of \(x\) in our original function \(f(x) = \sqrt[3]{x}\). So, we get \(f(f^{-1}(x)) = f(x^3) = \sqrt[3]{(x^3)} = x\). This verifies the first part (i.e., \(f(f^{-1}(x)) = x\)).

Step 3: Verification Part 2

The next verification is to verify that \(f^{-1}(f(x)) = x\). This means that we need to replace \(f(x)\) with its given form in \(f^{-1}(x)\). That is, we replace \(x\) in \(f^{-1}(x) = x^3\) with \(f(x) = \sqrt[3]{x}\). This gives us \(f^{-1}(f(x)) =(\sqrt[3]{x})^3 = x\). This verifies the second part.

Key Concepts

Understanding One-to-One Functions

A **one-to-one function** is a function where each output value corresponds to exactly one input value, and vice versa. This means that no two different input values will map to the same output value. Understanding this property is crucial when working with inverse functions because only one-to-one functions can have inverses that are also functions.

To determine if a function is one-to-one, you can apply the **Horizontal Line Test**: if any horizontal line crosses the graph of the function at most once, the function is one-to-one.

• For example, imagine the function graph of a strict diagonal line. Any horizontal line will intersect this line once, showing that it's one-to-one.
• An example of a function that is not one-to-one is a parabola, since horizontal lines can intersect it more than once.

One-to-one functions are an essential step in identifying whether you can find an inverse function, which will be detailed further below.

The Process of Function Verification

**Function verification** involves confirming that a proposed inverse function is indeed the correct inverse of the original function. This is done by ensuring that both the function composed with its inverse, and the inverse composed with the function, return the original input value.

For the function \( f(x) = \sqrt[3]{x} \), the inverse function is found as \( f^{-1}(x) = x^3 \). Verification requires showing:

• First verification step: \( f(f^{-1}(x)) = f(x^3) = \sqrt[3]{x^3} = x \)
• Second verification step: \( f^{-1}(f(x)) = (\sqrt[3]{x})^3 = x \)

Both steps conclude with \( x \), demonstrating that our inverse function is correct.
This process of verification is integral to ensure that the computations maintain the integrity of transformations across functions and inverses.

Exploring the Cubic Root Function

The **cubic root function** is a mathematical function represented as \( f(x) = \sqrt[3]{x} \), which takes any real number and returns its cubic root. Unlike square root functions, cubic root functions are one-to-one over the entire set of real numbers; this makes them excellent candidates for having inverses.

Here are a few interesting points about the cubic root function:

• It passes through the origin \((0, 0)\).
• It extends infinitely in both positive and negative directions since any number can have a cube root.
• Graphically, it exhibits rotational symmetry, allowing every real number to have a unique cubic root.

Understanding the behavior of the cubic root function helps in recognizing its inverse, \( f^{-1}(x) = x^3 \), can also handle all real numbers, reinforcing its usability in various mathematical transformations and proofs.