Finding The Inverse Of Y Equals X Squared Minus 10x

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Understanding inverse functions is a crucial concept in mathematics, particularly when dealing with quadratic equations. In this comprehensive guide, we will walk you through the process of finding the inverse of the quadratic function $y = x^2 - 10x$. We will break down each step, explain the underlying principles, and provide clear examples to ensure you grasp the concept thoroughly. This guide is designed to help students, educators, and anyone interested in deepening their understanding of inverse functions and quadratic equations.

1. Introduction to Inverse Functions

Inverse functions are a fundamental concept in mathematics, and understanding them is crucial for various applications, from solving equations to analyzing mathematical models. An inverse function essentially reverses the operation of the original function. To put it simply, if a function $f(x)$ takes an input $x$ and produces an output $y$, then its inverse function, denoted as $f^{-1}(x)$, takes $y$ as an input and returns $x$. This reversal property is the core idea behind inverse functions. For a function to have an inverse, it must be one-to-one, meaning that each input corresponds to a unique output, and each output corresponds to a unique input. This condition ensures that the reversal process is well-defined. Visually, a one-to-one function passes the horizontal line test, which means that no horizontal line intersects the graph of the function more than once. If a function is not one-to-one over its entire domain, we can sometimes restrict the domain to a subset where it is one-to-one, allowing us to find an inverse for that restricted domain.

Think of it this way: Imagine a machine that takes a number, say 3, performs some operations on it, and outputs a result, let's say 5. The inverse function is like a machine that takes the output 5 and reverses the operations to give you back the original input, 3. This reverse process is what makes inverse functions so useful. They allow us to "undo" the original function and solve for the input based on the output. This is particularly helpful in solving equations where we need to isolate a variable. In the context of graphs, the graph of an inverse function is a reflection of the original function across the line $y = x$. This means that if a point $(a, b)$ lies on the graph of the original function, then the point $(b, a)$ lies on the graph of the inverse function. Understanding this graphical relationship can provide valuable insights into the behavior of inverse functions.

2. Understanding the Given Quadratic Function: $y = x^2 - 10x$

The given quadratic function is $y = x^2 - 10x$. Before we dive into finding its inverse, let's first understand the properties of this quadratic function. Quadratic functions are polynomial functions of degree two, generally expressed in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if $a > 0$) or downwards (if $a < 0$). The vertex of the parabola is the point where the curve changes direction, representing either the minimum or maximum value of the function. For the given function $y = x^2 - 10x$, we can identify that $a = 1$, $b = -10$, and $c = 0$. Since $a = 1$ is positive, the parabola opens upwards, indicating that the function has a minimum value.

To better understand the function, we can complete the square to rewrite it in vertex form, which is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. Completing the square involves manipulating the quadratic expression to create a perfect square trinomial. For the given function, we have:

$y = x^2 - 10x$

To complete the square, we take half of the coefficient of the $x$ term (which is -10), square it ((-5)^2 = 25), and add and subtract it within the equation:

$y = x^2 - 10x + 25 - 25$

Now, we can rewrite the first three terms as a perfect square:

$y = (x - 5)^2 - 25$

From this vertex form, we can see that the vertex of the parabola is at the point $(5, -25)$. This means the minimum value of the function is -25, which occurs when $x = 5$. This information is crucial because it tells us about the range of the function and helps us determine if the function is one-to-one. Because the parabola opens upwards and has a vertex at $(5, -25)$, the function is not one-to-one over its entire domain (all real numbers). A horizontal line can intersect the parabola at two points, violating the horizontal line test. However, we can restrict the domain to either $x extgreater= 5$ or $x extless= 5$ to make the function one-to-one and thus invertible over that restricted domain.

3. Steps to Find the Inverse of $y = x^2 - 10x$

Finding the inverse of a function involves a systematic approach. Here are the steps we'll follow to find the inverse of $y = x^2 - 10x$.

Step 1: Replace $y$ with $x$ and $x$ with $y$

The first step in finding the inverse of a function is to interchange the roles of $x$ and $y$. This is because the inverse function essentially reverses the input-output relationship of the original function. So, wherever you see $y$ in the original equation, replace it with $x$, and wherever you see $x$, replace it with $y$. This step reflects the fundamental concept of an inverse function: it takes the output of the original function as its input and produces the original function's input as its output. For the given function $y = x^2 - 10x$, this step transforms the equation into:

$x = y^2 - 10y$

This new equation represents the inverse relationship between $x$ and $y$. Our goal now is to solve this equation for $y$, which will give us the expression for the inverse function. This interchange of variables is a crucial step and ensures that we are working with the inverse relationship.

Step 2: Solve for $y$

The next step is to solve the equation for $y$. This will give us the explicit form of the inverse function. From the previous step, we have the equation:

$x = y^2 - 10y$

This is a quadratic equation in terms of $y$. To solve for $y$, we need to rewrite the equation in a form that allows us to isolate $y$. One common method is to complete the square, which we previously used to analyze the original function. To complete the square for the equation $x = y^2 - 10y$, we follow a similar process. We take half of the coefficient of the $y$ term (which is -10), square it ((-5)^2 = 25), and add it to both sides of the equation. This maintains the equality while allowing us to form a perfect square trinomial:

$x + 25 = y^2 - 10y + 25$

Now, we can rewrite the right side of the equation as a perfect square:

$x + 25 = (y - 5)^2$

Next, we take the square root of both sides of the equation. It's important to remember that when taking the square root, we need to consider both the positive and negative roots:

$ ext{\pm\sqrt{x + 25}} = y - 5$

Finally, we isolate $y$ by adding 5 to both sides:

$y = 5 ext{\pm\sqrt{x + 25}}$

This equation gives us two possible solutions for $y$, reflecting the fact that the original quadratic function is not one-to-one over its entire domain. This is where the restriction of the domain becomes crucial, as we'll discuss in the next step.

Step 3: Consider the Domain Restriction

As we discussed earlier, the original function $y = x^2 - 10x$ is not one-to-one over its entire domain. This means that its inverse, as we found in the previous step ($y = 5 ext{\pm\sqrt{x + 25}}$), has two possible solutions for each input $x$. To define a proper inverse function, we need to restrict the domain of the original function so that it becomes one-to-one. This restriction will then determine which part of the inverse function we should consider.

Recall that the vertex of the parabola $y = x^2 - 10x$ is at $(5, -25)$. This vertex divides the parabola into two halves, each of which is one-to-one. We can restrict the domain of the original function to either $x extgreater= 5$ or $x extless= 5$. Let's consider each case:

1. If we restrict the domain to $x extgreater= 5$, the original function is increasing. In this case, the range of the original function is $y extgreater= -25$. The inverse function will then be the part of $y = 5 ext{\pm\sqrt{x + 25}}$ that is also increasing. Since we restricted $x$ to be greater than or equal to 5, we choose the positive square root to match this behavior:

   $y = 5 + ext{\sqrt{x + 25}}$, for $x extgreater= -25$

2. If we restrict the domain to $x extless= 5$, the original function is decreasing. In this case, the range of the original function is also $y extgreater= -25$. The inverse function will then be the part of $y = 5 ext{\pm\sqrt{x + 25}}$ that is also decreasing. Since we restricted $x$ to be less than or equal to 5, we choose the negative square root to match this behavior:

   $y = 5 - ext{\sqrt{x + 25}}$, for $x extgreater= -25$

The domain of the inverse function is determined by the range of the original function. Since the range of $y = x^2 - 10x$ (with either domain restriction) is $y extgreater= -25$, the domain of the inverse function is $x extgreater= -25$. This is because the inverse function takes the outputs of the original function as its inputs.

In summary, the inverse function depends on the domain restriction applied to the original function. Without a specific domain restriction, the inverse is technically $y = 5 ext{\pm\sqrt{x + 25}}$. However, to have a proper inverse function, we need to choose either the positive or negative square root based on the domain restriction.

4. Identifying the Correct Inverse Function

Now that we have derived the inverse function $y = 5 ext{\pm\sqrt{x + 25}}$, let's compare it with the given options to identify the correct answer. The options provided are:

• A. $y= ext{\pm\sqrt{x-25}}-5$
• B. $y= ext{\pm\sqrt{x-25}}+5$
• C. $y= ext{\pm\sqrt{x+25}}-5$
• D. $y= ext{\pm\sqrt{x+25}}+5$

Comparing our derived inverse function $y = 5 ext{\pm\sqrt{x + 25}}$ with the given options, we can see that option D, $y= ext{\pm\sqrt{x+25}}+5$, is the correct answer. This option matches the form we obtained after completing the square and solving for $y$.

It's crucial to note that the other options are incorrect. Option A and B have $x - 25$ inside the square root, which is not what we obtained when completing the square. Option C has the correct $x + 25$ term inside the square root but has a $-5$ outside the square root, which is also incorrect. Option D is the only one that correctly represents the inverse function we derived.

Therefore, the correct inverse function is:

$y= ext{\pm\sqrt{x+25}}+5$

This answer reflects the complete process of finding the inverse, including interchanging $x$ and $y$, completing the square, solving for $y$, and considering the $ ext{\pm}$ sign due to the square root.

5. Verification and Conclusion

To further verify our result, we can consider a specific example. Let's take a point on the original function and see if its inverse corresponds correctly on our derived inverse function. Consider $x = 6$ in the original function $y = x^2 - 10x$. Plugging in $x = 6$, we get:

$y = (6)^2 - 10(6) = 36 - 60 = -24$

So, the point $(6, -24)$ lies on the original function. Now, let's use the inverse function $y= ext{\pm\sqrt{x+25}}+5$ and plug in $x = -24$:

$y = ext{\pm\sqrt{-24 + 25}} + 5 = ext{\pm\sqrt{1}} + 5 = ext{\pm 1} + 5$

We have two possible solutions: $y = 1 + 5 = 6$ and $y = -1 + 5 = 4$. Since we considered $x = 6$ in the original function (where $x extgreater 5$), we should consider the positive square root in the inverse function. Thus, $y = 6$, which matches our original $x$ value. This confirms that our derived inverse function is correct.

In conclusion, we have successfully found the inverse of the quadratic function $y = x^2 - 10x$. The key steps involved interchanging $x$ and $y$, completing the square, solving for $y$, and considering the domain restriction. The correct inverse function is:

$y= ext{\pm\sqrt{x+25}}+5$

This process demonstrates the importance of understanding inverse functions and their properties, especially when dealing with non-one-to-one functions like quadratics. By restricting the domain, we can define a proper inverse function that reverses the operation of the original function over that restricted domain. Understanding these concepts is crucial for advanced mathematical studies and various applications in science and engineering.