Problem 14 - 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 \(f^{-1}(f(x))=x\) $$f(x)=4 x$$

Step-by-Step Solution

Answer

The inverse for the function \(f(x)=4x\) is \(f^{-1}(x) = x/4\), and it is verified by the conditions \(f(f^{-1}(x))=x\) and \(f^{-1}(f(x))=x\).

Step 1: Compute the Inverse of the function

To find the inverse of a function, typically you replace the function notation with \(y\), switch \(x\) and \(y\), and then solve for \(y\) to get the inverse function.\nSo, if \(f(x)=4x\), we can say that \(y=4x\).\nSwitch \(x\) and \(y\) we get \(x=4y\). Now solving for \(y\) (which will be our inverse function \(f^{-1}(x)\)), we find that \(f^{-1}(x)=x/4\).

Step 2: Verify the Equation - Part 1

We have obtained that the inverse function is \(f^{-1}(x)=x/4\). To verify this, we need to ensure that \(f(f^{-1}(x))=x\). Substituting \(f^{-1}(x)\) into \(f(x)\), we obtain \(f(f^{-1}(x)) = 4f^{-1}(x) = 4*(x/4) = x\). This verifies the first condition of the inverse function.

Step 3: Verify the Equation - Part 2

The second condition for the verification is that \(f^{-1}(f(x))=x\). Substituting \(f(x)\) into \(f^{-1}(x)\), we obtain \(f^{-1}(f(x)) = f^{-1}(4x) = (4x)/4 = x\). This verifies the second condition of the inverse function.

Key Concepts

One-to-One Functions

In mathematics, a function is considered one-to-one, or injective, if each input is associated with a unique output. This property is crucial for a function to have an inverse. Why is this important? Well, if a function is one-to-one, it means no two different input values will produce the same output value. This unique mapping ensures that we can "reverse" the function, essentially finding a unique input for every output. When checking if a function is one-to-one, you can use the Horizontal Line Test. By drawing horizontal lines across the graph of the function, you can see if any line intersects the graph more than once. If it does not, the function is one-to-one. For example, the linear function given in the original exercise, \(f(x) = 4x\), is a classic one-to-one function. Any horizontal line will intersect its graph only once, confirming its injectiveness. This means \(f(x) = 4x\) ensures an inverse function can exist, which is crucial for the subsequent steps in solving inverse problems.

Function Verification

Verifying a function is especially important when dealing with inverses. After computing an inverse function, it's crucial to confirm that the operation was done correctly. This involves ensuring the original function and its inverse "undo" each other. For a function \(f(x)\) and its inverse \(f^{-1}(x)\), two equations must hold true:

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

These equations confirm the function's and inverse's roles in perfectly reversing each other's operations. Let's break down these verifications for clarity.

Verification Part 1

By substituting \(f^{-1}(x)\) into \(f(x)\), you should retrieve the original input value \(x\). For our exercise's function, \(f(f^{-1}(x)) = 4(x/4) = x\) checks out correctly. This ensures that using the inverse of \(f(x)\) in \(f\) returns the original \(x\).

Verification Part 2

Similarly, substituting \(f(x)\) into the inverse function should also return \(x\). Thus, \(f^{-1}(f(x)) = (4x)/4 = x\) proves correct for our problem. Successfully demonstrating these two conditions confirms our calculations and solidifies understanding.

Solving Equations

Solving equations is a fundamental part of finding inverse functions. By switching input and output, we rearrange the equation to isolate the new output, leading us to the inverse function. Let's break down the process using our example function \(f(x) = 4x\). Initially, replace \(f(x)\) with \(y\), giving us \(y = 4x\). To solve for the inverse, switch \(x\) and \(y\) to reflect the swapping of input/output roles, obtaining \(x = 4y\). The next step focuses on isolating \(y\). For this problem, to solve for \(y\), divide both sides by 4: \(y = x/4\). This result, \(f^{-1}(x) = x/4\), represents the inverse function.

Key Tips for Solving

• Always carefully swap \(x\) and \(y\) when starting to find the inverse.
• Perform algebraic operations step by step. This prevents errors and ensures clarity.
• Verify your final result with substitution, as detailed in the function verification section, to ensure complete correctness.

Mastering this equation-solving technique is foundational for confidently dealing with various inverse problems.