Problem 11 - Precalculus Essentials

Question

The functions 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=x+3$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x + 3\) is \(f^{-1}(x) = x - 3\), and it has been verified that this is correct because both \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Step 1: Find the Inverse

Replace \(f(x)\) with \(y\) to get \(y = x + 3\). To find the inverse, one needs to swap \(x\) and \(y\) and solve the resulting equation for \(y\). Doing this, we get \(x = y + 3\). Solving the equation for \(y\) yields \(y = x - 3\). Therefore, \(f^{-1}(x) = x - 3\).

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

To verify, replace \(x\) in \(f(x)\) with the inverse function. So, we have \(f(f^{-1}(x)) = f(x - 3) = (x - 3) + 3 = x\). This simplifies to \(x\), showing that \(f(f^{-1}(x)) = x\).

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

The final verification involves replacing \(x\) in \(f^{-1}(x)\) with the original function to see if this also simplifies to \(x\). So, \(f^{-1}(f(x)) = f^{-1}(x+3) = (x + 3) - 3 = x\). This simplifies to \(x\), showing that \(f^{-1}(f(x)) = x\).

Key Concepts

One-to-One Functions

A fundamental concept in algebra and calculus is that of one-to-one functions. These functions, also known as injective functions, have a unique feature where each element of the range corresponds to exactly one element of the domain. In other words, no two different values in the domain map to the same value in the range.

This characteristic is crucial for the existence of inverse functions. If a function is not one-to-one, it cannot have an inverse that is a function. To visualize this, think about a function as a pairing process where no two inputs can be paired with the same output. For instance, if you attend a dance where each person must dance with exactly one partner, a one-to-one function would ensure that no one is left without a partner and no one has more than one partner.

Checking for One-to-Oneness

A simple way to check for this is by using the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one. Additionally, mathematical tests involving derivatives (for continuously differentiable functions) or difference quotients (for discrete functions) can also confirm one-to-oneness.

Verifying Inverse Functions

Once you suspect you have found an inverse function, you must verify it to guarantee that it undoes the effects of the original function. This is the heart of what makes an inverse function special: it reverses the input-output relationship of the original function.

Verification is a two-step process:

• First, you ensure that applying the original function to the inverse returns you to your starting value: if you start with any value 'x', substitute it into the inverse to get 'y', and then apply the original function to 'y', you should end up with 'x' again, symbolically represented as \( f(f^{-1}(x)) = x \).
• Second, you confirm that applying the inverse function to the original function also returns the starting value: if you start with any value 'x', apply the original function to get 'y', and then apply the inverse to 'y', you should return to 'x', which is expressed as \( f^{-1}(f(x)) = x \).

If both conditions are met, you have successfully verified that the two functions are indeed inverses of each other.

Finding Inverse Equations

To find the equation of an inverse function, there's a standard series of transformations you can rely on. Starting with the function you wish to invert, labeled as \(f(x)\):

The Process of Finding an Inverse

1. First, you replace \(f(x)\) with \(y\). This step is simply a notational change that makes the following steps clearer.
2. Next, you swap the roles of \(x\) and \(y\). This is the crux of inverting a function—what was the output ('y') is now the input ('x'), and vice versa.
3. After the swap, solve for the new \(y\). This usually involves algebraic manipulations to isolate \(y\) on one side of the equation.
4. Finally, rewrite the function with the new \(y\) as \(f^{-1}(x)\), which denotes the inverse function.

These steps are a mirrored version of the original function's process, which reflects the intrinsic property of inverses to reverse each other's actions. Once you've found the inverse, don't forget to review whether the function is indeed one-to-one, to ensure the inverse is valid across the entire domain of the original function.