Problem 14 - College Algebra

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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$ f(x)=4 x $$

Step-by-Step Solution

Answer

The inverse of \(f(x) = 4x\) is \(f^{-1}(x) = x/4\), and it satisfies \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). Therefore, it is indeed the correct inverse.

Step 1: Solve for the Inverse

To find the equation for \(f^{-1}(x)\), the inverse function, one should replace \(f(x)\) with \(y\) in the original function, so that it becomes \(y = 4x\). Afterward, swapping \(x\) and \(y\) to get \(x = 4y\), and then isolating \(y\) by dividing both sides by 4 gives \(y = x/4\). So, \(f^{-1}(x) = x/4\).

Step 2: Verification I: \(f(f^{-1}(x)) = x\)

Substitute \(f^{-1}(x)\) with \(x/4\) into the original function, yielding \(f(f^{-1}(x)) = 4(x/4) = x\). This verifies the first part of the verification process.

Step 3: Verification II: \(f^{-1}(f(x)) = x\)

Substitute \(f(x)\) with \(4x\) into the inverse function, yielding \(f^{-1}(f(x)) = (4x)/4 = x\). This verifies the second part of the verification process.

Key Concepts

One-to-One Functions

One-to-one functions, or injective functions, play a crucial role in understanding inverse functions. These functions have a special property: each input is mapped to a unique output. For example, in a one-to-one function like the linear function given, if you know the output, you can determine the input without any ambiguity.

This is a necessary condition for a function to have an inverse. If a function is not one-to-one, then its inverse would not be well-defined, because a single output could correspond to multiple inputs. This unique mapping ensures that when we attempt to reverse the function, with an inverse, every output leads back to exactly one input.

Function Composition

Function composition is key to verifying inverse functions. When composing functions, you essentially apply one function to the results of another. For the relationship between a function and its inverse, we have two important compositions:

• Firstly, applying the function to its inverse, represented as \(f(f^{-1}(x))\), should simply return \(x\).
• Secondly, applying the inverse to the function, indicated by \(f^{-1}(f(x))\), should also return \(x\). These formulations are the checks for the correctness of an inverse.

The composition acts like a mathematical proof, affirming that when a function and its inverse are composed, they cancel each other out, leading back to the original value.

Inverse Verification

Verifying inverses is an essential part of ensuring that the inverse function is correctly derived. In our problem, the function was \(f(x) = 4x\) and its inverse is \(f^{-1}(x) = x/4\). To verify:

• Check that \(f(f^{-1}(x)) = 4(x/4) = x\). This checks that the function when applied to its inverse returns the original \(x\), confirming our first verification.
• Then, verify \(f^{-1}(f(x)) = (4x)/4 = x\). This ensures that applying the inverse to the function also returns \(x\).

If both verifications hold true, the inverse function is validated as correct, reaffirming the core concept of inverses.

Algebraic Manipulation

Algebraic manipulation involves altering the structure of an equation while keeping its integrity intact. It is a fundamental skill for finding the inverse of a function. Let's look at the process:

• Begin by setting the original function, \(f(x) = 4x\), equal to \(y\), so that it becomes \(y = 4x\). This is a simple re-arrangement that sets the stage for our next step.
• Swap \(x\) and \(y\) to reverse roles, resulting in \(x = 4y\). This swap is pivotal as it transforms our viewpoint, approaching the equation from the perspective of outputs mapping back to inputs.
• Finally, solve for \(y\) through algebraic manipulation, giving you \(y = x/4\). Isolating \(y\) shows us the inverse, \(f^{-1}(x)\).

These steps demonstrate how algebraic techniques uncover the inverse, leveraging properties of equations with precision.