Finding The Inverse Of Y = X² - 12 A Comprehensive Guide

Finding the inverse of a function is a fundamental concept in mathematics, particularly in algebra and calculus. The inverse function essentially "undoes" the original function. In simpler terms, if a function ${ f }$ maps ${ x }$ to ${ y }$, then the inverse function, denoted as ${ f^{-1} }$, maps ${ y }$ back to ${ x }$. In this article, we will delve into the process of finding the inverse of the function ${ y = x^2 - 12 }$ and discuss the implications and nuances involved.

Understanding Inverse Functions

Before we dive into the specifics, let's clarify what an inverse function is and why it is important. A function ${ f^{-1} }$ is the inverse of ${ f }$ if and only if ${ f^{-1}(f(x)) = x }$ for all ${ x }$ in the domain of ${ f }$ and ${ f(f^{-1}(y)) = y }$ for all ${ y }$ in the range of ${ f }$. Not all functions have inverses. For a function to have an inverse, it must be bijective, meaning it is both injective (one-to-one) and surjective (onto).

• Injective (One-to-One): A function is injective if each element of the range is associated with at most one element of the domain. Graphically, a function is injective if it passes the horizontal line test, meaning no horizontal line intersects the graph more than once.
• Surjective (Onto): A function is surjective if every element of the range is associated with at least one element of the domain. In other words, the function's range is equal to its codomain.

The function ${ y = x^2 - 12 }$ is a quadratic function, and quadratic functions are not injective over their entire domain (all real numbers) because they produce the same y-value for both positive and negative x-values (e.g., ${ (-x)^2 = x^2 }$). However, we can restrict the domain to make the function injective and thus invertible.

Steps to Find the Inverse

To find the inverse of a function, we typically follow these steps:

1. Replace ${ f(x) }$ with ${ y }$.
2. Swap ${ x }$ and ${ y }$.
3. Solve for ${ y }$.
4. Replace ${ y }$ with ${ f^{-1}(x) }$.

Let's apply these steps to the function ${ y = x^2 - 12 }$.

Step 1: Replace ${ f(x) }$ with ${ y }$

Our function is already in this form: ${ y = x^2 - 12 }$.

Step 2: Swap ${ x }$ and ${ y }$

Swapping ${ x }$ and ${ y }$ gives us:

${ x = y^2 - 12 }$

Step 3: Solve for ${ y }$

Now, we need to isolate ${ y }$ in the equation. First, add 12 to both sides:

${ x + 12 = y^2 }$

Next, take the square root of both sides:

${ y = \pm \sqrt{x + 12} }$

It’s crucial to understand why we have a ${ \pm }$ (plus or minus) sign here. When we take the square root of a number, we must consider both the positive and negative roots because both ${ (\sqrt{x+12})^2 }$ and ${ (-\sqrt{x+12})^2 }$ equal ${ x + 12 }$. This is a direct consequence of the property that squaring a negative number yields a positive result.

Step 4: Replace ${ y }$ with ${ f^{-1}(x) }$

Finally, we express the inverse function as:

${ f^{-1}(x) = \pm \sqrt{x + 12} }$

Thus, the inverse of the function ${ y = x^2 - 12 }$ is ${ y = \pm \sqrt{x + 12} }$. This result indicates that for each value of ${ x }$ greater than or equal to -12, there are two possible values of ${ y }$ in the inverse function, corresponding to the positive and negative square roots.

Analyzing the Inverse Function

The inverse function ${ f^{-1}(x) = \pm \sqrt{x + 12} }$ is defined for ${ x \geq -12 }$ because the expression inside the square root must be non-negative. The ${ \pm }$ sign indicates that the inverse is not a function over its entire domain unless we restrict the range. This is because a function must have a unique output for each input. To make the inverse a function, we can choose either the positive or the negative square root.

Restricting the Domain

To obtain a true inverse function, we typically restrict the domain of the original function ${ y = x^2 - 12 }$. If we restrict the domain of the original function to ${ x \geq 0 }$, then the inverse function is:

${ f^{-1}(x) = \sqrt{x + 12} }$

If we restrict the domain of the original function to ${ x \leq 0 }$, then the inverse function is:

${ f^{-1}(x) = -\sqrt{x + 12} }$

These restrictions ensure that the inverse is a function, as each ${ x }$ value maps to a unique ${ y }$ value.

Graphical Interpretation

The graph of the inverse function is a reflection of the original function across the line ${ y = x }$. For the function ${ y = x^2 - 12 }$, the graph is a parabola opening upwards with its vertex at ${ (0, -12) }$. The inverse function ${ y = \pm \sqrt{x + 12} }$ represents a sideways parabola opening to the right, with its vertex at ${ (-12, 0) }$. The reflection across ${ y = x }$ interchanges the x and y coordinates of the key features of the graph, such as the vertex.

When we restrict the domain of the original function, we essentially take only one half of the parabola. For example, restricting to ${ x \geq 0 }$ means we only consider the right half of the parabola, and its reflection across ${ y = x }$ is the upper half of the sideways parabola, corresponding to ${ f^{-1}(x) = \sqrt{x + 12} }$.

Common Mistakes and Pitfalls

When finding inverse functions, several common mistakes can occur. Being aware of these pitfalls can help prevent errors:

1. Forgetting the ${ \pm }$ Sign: A frequent error is forgetting to include both positive and negative roots when taking the square root. Remember that both positive and negative values, when squared, can yield the same result.
2. Not Restricting the Domain: Failing to restrict the domain when the original function is not injective can lead to an inverse that is not a function. It’s crucial to determine an appropriate domain restriction to ensure the inverse has a unique output for each input.
3. Incorrectly Swapping Variables: Sometimes, individuals make mistakes when swapping ${ x }$ and ${ y }$. Ensure that you correctly interchange the variables before solving for ${ y }$.
4. Misunderstanding the Definition of an Inverse: A clear understanding of what an inverse function represents is essential. It “undoes” the original function, so applying ${ f }$ and then ${ f^{-1} }$ (or vice versa) should return the original input.
5. Algebraic Errors: Simple algebraic mistakes, such as errors in isolating ${ y }$, can lead to an incorrect inverse function. Always double-check each step to minimize errors.

Practical Applications of Inverse Functions

Inverse functions are not just theoretical constructs; they have practical applications in various fields:

1. Cryptography: In cryptography, inverse functions are used in encryption and decryption processes. Encryption functions transform plaintext into ciphertext, and the inverse function decrypts the ciphertext back into plaintext.
2. Computer Graphics: In computer graphics, transformations such as rotations and scaling are represented by matrices. The inverse matrix corresponds to the inverse transformation, allowing objects to be transformed back to their original state.
3. Calculus: In calculus, inverse functions are essential for finding antiderivatives and solving differential equations. The inverse trigonometric functions, for example, are crucial for integrating certain types of functions.
4. Economics: In economics, inverse demand and supply functions are used to analyze market equilibrium. The demand function relates the quantity of a good that consumers are willing to buy to its price, and the inverse demand function expresses the price as a function of the quantity demanded.
5. Engineering: In various engineering disciplines, inverse functions are used to solve problems involving system design and control. For example, in control systems, the inverse transfer function can be used to determine the input required to achieve a desired output.

Conclusion

Finding the inverse of the function ${ y = x^2 - 12 }$ involves swapping ${ x }$ and ${ y }$, solving for ${ y }$, and considering the ${ \pm }$ sign when taking square roots. The inverse function is ${ y = \pm \sqrt{x + 12} }$. However, to ensure the inverse is a function, we typically restrict the domain of the original function. Understanding the concept of inverse functions is crucial in mathematics and has practical applications in various fields, including cryptography, computer graphics, calculus, economics, and engineering. By carefully following the steps and being mindful of potential pitfalls, you can confidently find and analyze inverse functions.

The correct answer to the question is:

B. ${ y = \pm \sqrt{x + 12} }$