which equation is the inverse of y x2 16

Daniel Parker

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22-02-2025

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The question of finding the inverse of the equation y = x² + 16 is a common one in algebra. Understanding how to find the inverse of a function is crucial for many mathematical applications. Let's break down how to solve this problem step-by-step.

Understanding Inverse Functions

Before we jump into the specific equation, let's clarify what an inverse function is. An inverse function essentially "undoes" what the original function does. If you apply a function and then its inverse, you end up back where you started. Mathematically, if f(x) is a function and f⁻¹(x) is its inverse, then f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Finding the Inverse of y = x² + 16

Now, let's tackle the equation y = x² + 16. To find its inverse, we follow these steps:

1. Swap x and y: This is the first crucial step. We replace every instance of 'y' with 'x' and every 'x' with 'y'. This gives us: x = y² + 16.

2. Solve for y: Now, we need to isolate 'y' in the equation. This involves some algebraic manipulation:

   • Subtract 16 from both sides: x - 16 = y²
   • Take the square root of both sides: ±√(x - 16) = y

3. The Inverse Function: Therefore, the inverse function is y = ±√(x - 16). Notice the ± sign. This is important because the square root of a number can be both positive and negative.

Important Considerations: The ± Sign and Restrictions

The ± sign in the inverse function, y = ±√(x - 16), indicates that for any given x value (except for x=16), there are two corresponding y values. This is a consequence of the original function, y = x² + 16, not being a one-to-one function. A one-to-one function means that each x value maps to only one unique y value, and vice versa. The original function fails this criterion because both x=4 and x=-4 produce the same y-value (y=32).

To make the inverse a proper function (meaning one output for each input), we often need to restrict the domain of the original function. For instance, if we restrict the domain of y = x² + 16 to x ≥ 0, then the inverse becomes y = √(x - 16). Similarly, restricting the domain to x ≤ 0 would lead to y = -√(x - 16).

Graphing the Original and Inverse Functions

Graphing both the original function (y = x² + 16) and its inverse (y = ±√(x - 16)) visually demonstrates their relationship. You'll notice they are reflections of each other across the line y = x. This is a characteristic property of inverse functions.

In Summary

The inverse of y = x² + 16 is y = ±√(x - 16). However, remember the importance of the ± sign and the potential need to restrict the domain of the original function to obtain a true one-to-one inverse function. Understanding this nuance is key to mastering inverse functions.