Problem 14 - 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=4 x$$

Step-by-Step Solution

Answer

The inverse function for \( f(x) = 4x \) is \( f^{-1}(x) = x/4 \). The verification process confirms this is correct.

Step 1: Determine the Inverse Function

Initially, let's re-write the function \( f(x) = 4x \) as \( y = 4x \). In order to get the inverse function, swap \( y \) and \( x \), resulting in \( x = 4y \). Subsequently, solve for \( y \) to get \( y = x/4 \). Hence, the inverse of the function \( f(x) = 4x \) is \( f^{-1}(x) = x/4 \).

Step 2: Verify the Inverse Function

To verify the inverse function, we need to prove these two equations: \( f(f^{-1}(x))=x \) and \( f^{-1}(f(x))=x \). a) Substitute \( f^{-1}(x) = x/4 \) into \( f(x) = 4x \): \( f(f^{-1}(x)) = 4 * (x/4) = x \). b) Substitute \( f(x) = 4x \) into \( f^{-1}(x) = x/4 \): \( f^{-1}(f(x)) = (4x) / 4 = x \). Both equations equate to \( x \), so the inverse function is verified successfully.

Key Concepts

Understanding One-to-One Functions

In the world of mathematics, a one-to-one function, also known as an injective function, is a fundamental concept that is critical to understand before diving into inverse functions. A function is considered one-to-one if each element of the domain (the set of all possible inputs) corresponds to a unique element of the codomain (the set of all potential outputs). In simpler terms, if you put different inputs into the function, you'll get different outputs.

Visualizing this on a graph bears the Horizontal Line Test: if no horizontal line intersects the graph of the function at more than one point, then the function is one-to-one. This uniqueness is essential for the existence of an inverse function, because if a function had the same output for two different inputs, it would be impossible to determine a unique inverse. For the exercise given, the function defined by the equation \( f(x) = 4x \) is indeed one-to-one since each input value is multiplied by 4 to produce a unique output value.

Finding Inverse Functions Step by Step

Discovering the inverse of a function is akin to retracing your steps in a maze to find your way back to the start. For the function \( f(x) = 4x \), we initiated this journey by rewriting the function in terms of \( y \): \( y = 4x \).

Here's the tricky part: we swap \( x \) and \( y \), leading to the equation \( x = 4y \). It's like saying, 'If initially I multiply my input by 4 to get my output, what should I do to my output to recover my input?' The act of swapping hints at the 'undoing' process. From here, we solve for the new \( y \), which gives us the inverse function, \( f^{-1}(x) = x/4 \). The division by 4 'undoes' the multiplication by 4 in the original function. This step-by-step solution helps us to transition from our starting point to finding the path backwards.

Verifying Inverse Functions Reliably

Just as it's important to proofread your writing, it is crucial to verify that the inverse function you've found actually reflects the original function's actions in reverse. The verification process involves two checks carried out by the substitution method - essentially asking, 'Does my inverse function take me back to where I started?'

When we plug the inverse function \( f^{-1}(x) = x/4 \) into the original function \( f(x) = 4x \), we end up with \( f(f^{-1}(x)) = 4 * (x/4) = x \). Similarly, feeding the output of the original function back into its inverse, \( f^{-1}(f(x)) = (4x) / 4 = x \), also simplifies to \( x \). This twofold successful substitution is the mathematical equivalent of nodding in agreement—both operations confirm that the inverse function is indeed correct. It's a reassuring conclusion to a process that requires both precision and a firm grasp of the concepts.