Problem 22 - 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 \(f^{-1}(f(x))=x\) $$f(x)=\frac{2}{x}$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x)=\frac{2}{x}\). Verification shows that both \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\), respectively, which verifies that \(f^{-1}(x)\) is indeed the inverse function of \(f(x)\).

Step 1: Find the Inverse

To find the inverse of a function, one must first replace the function notation \(f(x)\) by \(y\). This gives:\n \(y=\frac{2}{x}\). Now, switch \(x\) and \(y\). This results in: \(x=\frac{2}{y}\). Solving for \(y\) gives the inverse function. So, multiply both sides by \(y\) and then divide both sides by \(x\) to get \(y=\frac{2}{x}\). Thus, the inverse function \(f^{-1}(x)\) is \(f^{-1}(x)=\frac{2}{x}\).

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

To verify the first part of the inverse, substitute \(f^{-1}(x)\) into the function \(f(x)\). So, \(f(f^{-1}(x))=f(\frac{2}{x})=\frac{2}{(2/x)} = x\). Hence, \(f(f^{-1}(x))=x\) is verified.

Step 3: Verify Inverse 2: \(f^{-1}(f(x))=x\)

To verify the second part of the inverse, substitute \(f(x)\) into \(f^{-1}(x)\). So, \(f^{-1}(f(x))=f^{-1}(\frac{2}{x})=\frac{2}{(2/x)} = x\). Hence, \(f^{-1}(f(x))=x\) is verified as well.

Key Concepts

One-to-One Functions

Before diving into inverse functions, it's crucial to grasp what a one-to-one function is. In essence, a one-to-one function ensures that every unique input corresponds to a unique output. This means no two different inputs will have the same output.
Such functions are vital for creating inverses because if a function wasn't one-to-one, it would be impossible to determine a clear reverse path from output back to input.
To check if a function is one-to-one, you can use the horizontal line test visually on a graph. If no horizontal line intersects the function’s graph more than once, it’s one-to-one. This property is key in ensuring that an inverse exists and is well-defined.

Verifying Inverses

Verifying that two functions are inverses of each other is an essential step. To prove that a function has correctly derived its inverse, two main checks are performed:

• Show that the composition of the function and its inverse, in both possible orders, returns the original input.
• The first check is to calculate \(f(f^{-1}(x))\) and ensure that it simplifies to \(x\).
• The second check is to calculate \(f^{-1}(f(x))\) and verify that it also simplifies to \(x\).

By performing these verifications, you're not just delving into algebraic manipulations, but also uncovering the fundamental relationship of input and output between a function and its reverse path.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging equations to solve for a desired variable. When finding an inverse function, you'll swap the roles of x and y in the original equation and then manipulate the equation to solve for y again. This process involves:

• Replacing \(f(x)\) with \(y\) to start with the function equation.
• Swapping \(x\) and \(y\), reflecting the idea of reversing the function.
• Using inverse operations like multiplying, dividing, adding, or subtracting to isolate \(y\).

These steps enable us to find \(f^{-1}(x)\) when given \(f(x)\). Mastering these techniques helps in understanding more complex mathematical problems and aids in recognizing patterns in data transformation.

Function Composition

Function composition involves combining two functions in a specific order where the output of one becomes the input of the other. This is denoted as \(f(g(x))\) or \((f \circ g)(x)\).
In the context of inverses, function composition is used to verify their correctness. If \(g(x)\) is the inverse of \(f(x)\):

• The operation \(f(g(x))\) should yield \(x\), as should \(g(f(x))\).

This symmetry illustrates how functions can undo each other’s work, bringing inputs back to their original state. Function composition is not only pivotal for inverses but also serves as a cornerstone in building advanced mathematical models.