Which Equation Is The Inverse Of Y 16x2 1

Finding the Inverse of y = 16x² + 1: A Comprehensive Guide

Finding the inverse of a function is a fundamental concept in algebra and calculus. It involves switching the roles of the independent and dependent variables (x and y) and solving for the new dependent variable. While straightforward for some functions, others, like the quadratic function y = 16x² + 1, present a slightly more complex challenge. This article will delve into the process of finding the inverse of y = 16x² + 1, exploring the nuances and potential complications along the way. We'll also discuss the characteristics of the original function and its inverse, and examine the implications for their respective graphs.

Understanding the Concept of Inverse Functions

Before we embark on the process of finding the inverse, let's solidify our understanding of what an inverse function actually is. An inverse function, denoted as f⁻¹(x), essentially "undoes" the operation performed by the original function, f(x). If we apply a function to a value and then apply its inverse to the result, we should obtain the original value. Mathematically, this can be expressed as:

f⁻¹(f(x)) = x and f(f⁻¹(x)) = x

This relationship holds true only if the original function is one-to-one (meaning each input has a unique output). The function y = 16x² + 1 is not one-to-one because both positive and negative values of x will produce the same positive y value. For example, x=1 and x=-1 both give y=17. To find an inverse function, we need to restrict the domain of the original function to make it one-to-one.

Step-by-Step Process: Finding the Inverse of y = 16x² + 1

To find the inverse, we follow these steps:

1. Swap x and y: This is the first crucial step in finding the inverse. We replace every instance of 'y' with 'x' and every 'x' with 'y'. This gives us:

   x = 16y² + 1

2. Solve for y: This is where the complexity lies for this particular quadratic function. Our goal is to isolate 'y' on one side of the equation. Let's proceed:

   x - 1 = 16y²

   (x - 1) / 16 = y²

   y = ±√((x - 1) / 16)

   y = ±(1/4)√(x - 1)

This equation represents two functions because of the ± sign. This highlights the fact that the original function is not one-to-one. To have a true inverse function, we need to restrict the domain of the original function.

3. Restricting the Domain: Since the original function y = 16x² + 1 is a parabola opening upwards, we can restrict its domain to either x ≥ 0 or x ≤ 0 to make it one-to-one. Let's choose x ≥ 0. This means the range of the inverse function will be y ≥ 0. Therefore, our inverse function becomes:

   y = (1/4)√(x - 1) for x ≥ 1 and y ≥ 0

This is the inverse function for the restricted domain of the original function. Note that we've chosen the positive square root because we've restricted the domain of the original function to non-negative values of x.

Analyzing the Original Function and its Inverse

Original Function: y = 16x² + 1

• Domain: All real numbers (-∞, ∞)
• Range: y ≥ 1 (all real numbers greater than or equal to 1)
• Graph: A parabola opening upwards with its vertex at (0, 1).

Inverse Function: y = (1/4)√(x - 1) (with restricted domain and range)

• Domain: x ≥ 1
• Range: y ≥ 0
• Graph: The graph of the inverse function is a portion of the square root function, shifted one unit to the right. It's a reflection of the restricted portion of the original parabola across the line y = x.

Verifying the Inverse

To verify that we've correctly found the inverse, we can perform the composition of functions:

• f(f⁻¹(x)): This means plugging the inverse function into the original function:

  f(f⁻¹(x)) = 16[(1/4)√(x - 1)]² + 1 = 16(1/16)(x - 1) + 1 = x - 1 + 1 = x

• f⁻¹(f(x)): This means plugging the original function into the inverse function:

  f⁻¹(f(x)) = (1/4)√(16x² + 1 - 1) = (1/4)√(16x²) = (1/4)(4|x|) = |x|

Since we restricted the domain of the original function to x ≥ 0, |x| = x. Therefore, f⁻¹(f(x)) = x. This confirms that our inverse function is correct for the restricted domain.

Implications for Graphing

The graph of the inverse function is a reflection of the graph of the original function (with the restricted domain) about the line y = x. This is a key characteristic of inverse functions. You'll observe that the restricted portion of the original parabola and its inverse are mirror images across this line.

Handling the Case with x ≤ 0

If we had chosen to restrict the domain of the original function to x ≤ 0 instead of x ≥ 0, the inverse function would have been:

y = -(1/4)√(x - 1) for x ≥ 1 and y ≤ 0

This represents the other half of the inverse relationship reflected across the y=x line. The choice of restriction influences which part of the inverse function you obtain.

Conclusion: Inverse Functions and Domain Restrictions

Finding the inverse of y = 16x² + 1 requires a careful understanding of inverse function concepts and the importance of domain restrictions. The original quadratic function is not one-to-one, meaning a direct inverse isn't possible without restricting its domain. By restricting the domain to either x ≥ 0 or x ≤ 0, we can derive a valid inverse function. Remember, the choice of restriction dictates the specific form of the inverse function obtained. This comprehensive explanation helps in understanding the detailed steps involved and highlights the critical role of domain restrictions in defining a proper inverse function for a non-one-to-one relation. The verification process and graphical analysis further solidify the understanding of the relationship between a function and its inverse. The article also covers the impact of choosing different domain restrictions and their effects on the inverse function's form. This provides a complete picture of the concept and its implications.