Problem 11 - 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=x+3$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x+3\) is \(f^{-1}(x) = x-3\) and it is verified by the two identities \(f(f^{-1}(x)) = x\ and\ f^{-1}(f(x)) = x\)

Step 1: Find the inverse function

Replace the function notation \(f(x)\) with \(y\). So, \(y = x+3\). Now, swap \(x\) and \(y\) to get the inverse function, getting \(x = y+3\). To solve for \(y\), subtract 3 from both sides of the equation to isolate \(y\), thus \(y = x - 3\). Consequently, the inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = x - 3\).

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

Substitute \(f^{-1}(x)\) in place of \(x\) in the equation \(f(x) = x+3.\) Therefore \(f(f^{-1}(x)) = f^{-1}(x) + 3 = (x - 3) + 3 = x\). Thus, this identity holds true.

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

Substitute \(f(x)\) in place of \(x\) in the equation \(f^{-1}(x) = x - 3.\) Therefore \(f^{-1}(f(x)) = f(x) - 3 = (x + 3) - 3 = x\). Thus, the inverse function satisfies both the identities thereby confirming it to be correctly obtained.

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each input value maps to a unique output value. This characteristic is crucial because it guarantees that the function has an inverse. A function being one-to-one is also known as being injective.

To identify a one-to-one function, you can use the Horizontal Line Test. If any horizontal line crosses the graph of the function at most once, the function is one-to-one. Alternatively, if you have the function as a formula, you should ensure that different inputs always result in different outputs.

For example, with the given function \(f(x) = x + 3\), every input \(x\) corresponds to a unique output \(x + 3\). Thus, \(f(x)\) is one-to-one, meaning it has an inverse function. This is the initial step needed to find the inverse because non-one-to-one functions do not have inverses.

Function Notation

Function notation provides a formal way to write functions for better clarity and understanding. The typical notation \(f(x)\) indicates a function \(f\) having \(x\) as its variable. This not only helps in quickly recognizing what the function is and its variable but also sets the groundwork for discussing inverse functions.

In the given solution, the notation \(f(x) = x + 3\) is used to describe the function. This notation tells us that for each input \(x\), the function produces an output by adding 3 to \(x\).

When dealing with inverse functions, you use notation like \(f^{-1}(x)\). This represents the inverse function which, when composed with the original function \(f\), returns the input number. Correct notation helps in making mathematical communication clear, especially when verifying the relationship between a function and its inverse.

Verification of Inverses

Verification of inverses is a critical concept that involves checking whether the functions truly are inverses of each other. This is done through composition of functions.

The verification process checks two conditions:

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

These conditions confirm that applying a function followed by its inverse (or vice versa) results in the original value.

Using our example, with \(f(x) = x + 3\) and \(f^{-1}(x) = x - 3\):
- Verify \(f(f^{-1}(x)) = (x - 3) + 3 = x\); it holds true.
- Verify \(f^{-1}(f(x)) = (x + 3) - 3 = x\); it also holds true.

Both statements being true confirms that \(f(x)\) and \(f^{-1}(x)\) are indeed inverses of each other.

Equation Solving

Equation solving is a fundamental step in finding the inverse of a function. When finding an inverse function, you essentially reverse the operations in the original function.

For example, given \(f(x) = x + 3\), you first swap \(x\) and \(y\) in \(y = x + 3\) to obtain \(x = y + 3\). Then solve for \(y\) by isolating it on one side of the equation:

• Subtract 3 from both sides: \(x - 3 = y\)
• Therefore, the inverse function is \(f^{-1}(x) = x - 3\)

By following these steps, you are applying algebraic manipulation to effectively "undo" the operation of the original function, allowing you to determine its inverse.

Solving equations accurately is key, as mistakes here can lead to incorrect results in finding and verifying inverse functions.