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

Step-by-Step Solution

Answer

The inverse function \(f^{-1}(x)\) is \(x - 3\). This has been verified correctly as both \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) equal to \(x\).

Step 1: Find the inverse

Interchange the x and y values in the given function. This gives \(x=y+3\). By solving for y, we can find the inverse function. This involves subtracting 3 from both sides which yields \(y = x - 3\). So, the inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = x - 3\).

Step 2: Verify the inverse

To verify the solution, substitute the reverse function into the original function and check if it meets the conditions for an inverse. This is done by showing that \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\). By substituting \(f^{-1}(x) = x - 3\) into \(f(x)\), we get \(f(f^{-1}(x)) = (x - 3) + 3 = x\). Likewise, by substituting \(f(x) = x + 3\) into \(f^{-1}(x)\), we get \(f^{-1}(f(x)) = (x + 3) - 3 = x\). So, the inverse has been verified correctly as both equal to \(x\).

Key Concepts

One-to-One Function

Understanding the one-to-one functions, also known as injective functions, is crucial for working with inverse functions. A one-to-one function has the unique property that each element of the domain is paired with a unique element in the range. No two different inputs in a one-to-one function lead to the same output. This exclusivity allows for each input to have an inverse output.

Let's visualize this with an example: Consider the function \( f(x) = x + 3 \). If you select any two distinct numbers, say \( a \) and \( b \), and apply them to the function, the resulting outputs \( f(a) \) and \( f(b) \) are different as long as \( a eq b \), thus confirming it's a one-to-one function. For the inverse of this function to exist, this condition must be met. By confirming the one-to-one nature of \( f(x) \), we can proceed to find its inverse with confidence.

Verifying Inverse Function

Once an inverse function, denoted as \( f^{-1}(x) \), is proposed, we need to verify it to ensure it meets the necessary criteria for being a true inverse. The technical aspect of this verification process involves two critical algebraic checks: \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \).

In our example, after finding the inverse to be \( f^{-1}(x) = x - 3 \), we verify it by plugging this into the original function: \( f(f^{-1}(x)) = (x - 3) + 3 = x \). Similarly, we confirm by inputting the original function into the inverse: \( f^{-1}(f(x)) = (x + 3) - 3 = x \). If both compositions result in the identity function, meaning that applying both functions successively brings you back to your original value, then the proposed inverse function is correct. This two-way verification builds a solid foundation for understanding inverse functions.

Algebraic Manipulation

Mastering algebraic manipulation is a must when working with functions and their inverses; it is the toolkit that allows us to rearrange equations and solve for unknowns. When finding the inverse of a function, we start by interchanging the \( x \) and \( y \) in the original equation, then solve for \( y \) to get the inverse function. This process often involves a series of algebraic steps such as adding, subtracting, multiplying, dividing, and factoring.

Inverting our function \( f(x) = x + 3 \) required us to perform a simple algebraic manipulation: subtracting 3 from both sides to isolate \( y \). The algebraic steps you take can vary in complexity depending on the form of the original function, but the goal remains the same: to algebraically 'undo' the original function, reflecting the core idea of an inverse – reversing the process.