Problem 13 - College Algebra

Question

The functions in Exercises \(11-30\) 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=2 x$$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=2x\) is \(f^{-1}(x)=\frac{x}{2}\).

Step 1: Find the Inverse

To find the inverse of the function \(f(x)=2x\), switch \(x\) and \(y\) to get \(x=2y\). Then solve for \(y\), \[y = \frac{x}{2}\]. This gives the inverse function, \(f^{-1}(x)=\frac{x}{2}\].

Step 2: Verify the Inverse - Part 1

First, verify the equation \(f(f^{-1}(x))=x\). Substitute \(f^{-1}(x)\) into the function \(f(x)\). \[f(f^{-1}(x)) = f(\frac{x}{2}) = 2(\frac{x}{2}) = x\], So, this proves the first equation.

Step 3: Verify the Inverse - Part 2

Second, verify the equation \(f^{-1}(f(x))=x\). Substitute \(f(x)\) into the function \(f^{-1}(x)\). \[f^{-1}(f(x)) = f^{-1}(2x) = \frac{2x}{2} = x\]. So, this proves the second equation. Thus, the function \(f^{-1}(x)=\frac{x}{2}\) is indeed the inverse of the function \(f(x) = 2x\).

Key Concepts

Understanding One-to-One Functions

A one-to-one function, also known as an injective function, is a special type of function where each output value corresponds to exactly one input value. This means that no two different input values map to the same output value.
In more technical terms, a function is one-to-one if, whenever \(f(a) = f(b)\), then \(a = b\). Understanding this is crucial because only one-to-one functions have inverses that are also functions.
This property is what allows us to "reverse" the function to find its inverse. In simple terms, if you can draw a horizontal line anywhere on the graph of a function and it only crosses the graphed function once, then the function is one-to-one. Verifying if a function is one-to-one can be important. This is often done by ensuring the function passes the horizontal line test or by checking if \(f(a) = f(b)\) leads to \(a = b\).

Verification of Inverse Functions

Verification of an inverse function is a critical step to ensure that the inverse we've found truly works as expected. For a function \(f(x)\) and its proposed inverse \(f^{-1}(x)\), we verify by checking two conditions:

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

In both cases, these equalities must hold true for the inverse to be correct.
In simpler terms, applying the original function to its inverse (or vice versa) should bring us back to our starting value \(x\).
Take the function \(f(x) = 2x\) and its inverse \(f^{-1}(x) = \frac{x}{2}\). Substituting \(f^{-1}(x)\) into \(f(x)\) gives us \(f\left(\frac{x}{2}\right)\) which simplifies to \(x\). Similarly, substituting \(f(x)\) into \(f^{-1}(x)\) results in \(\frac{2x}{2}\), which again equals \(x\).
Satisfying these conditions confirms our inverse is correct.

Algebraic Manipulation to Find Inverse

Finding the inverse of a function involves some algebraic manipulation. Let's break down the process into simple steps using the example \(f(x) = 2x\):

• Start with the equation \(f(x) = y\), which becomes \(2x = y\).
• Switch the roles of \(x\) and \(y\), giving us \(x = 2y\).
• Solve for \(y\) to find the inverse: \(y = \frac{x}{2}\).

This final expression for \(y\) is our inverse function \(f^{-1}(x) = \frac{x}{2}\).
Algebraic manipulation might involve steps such as switching variables and solving linear equations. Often, it requires care to correctly rearrange terms to isolate the variable of interest.
In each step, simplicity and accuracy are crucial to ensuring the inversion accurately reflects the original function. While this example is straightforward, complex functions might require additional operations such as completing the square or rationalizing expressions. Understanding these manipulations is essential for working with more advanced functions.