Finding The Inverse Of Y=5x^2+10 A Step By Step Guide

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In the realm of mathematics, understanding inverse functions is crucial, especially when dealing with quadratic equations. An inverse function, in essence, reverses the operation of the original function. This article will delve into the process of finding the inverse of a specific quadratic function, y = 5x² + 10, and guide you through the steps to simplify the equation effectively. This is a fundamental concept in algebra and calculus, often encountered in various mathematical and scientific contexts.

Understanding Inverse Functions

Before diving into the specifics, let's solidify the concept of inverse functions. If a function f takes an input x and produces an output y, its inverse function, denoted as f⁻¹*, takes y as input and returns the original x. In simpler terms, it undoes what the original function did. Mathematically, if f(x) = y, then f⁻¹(y) = x. This relationship is the cornerstone of finding inverse functions.

To find the inverse of a function, the general approach involves swapping the roles of x and y and then solving for y. This process essentially reverses the input-output relationship defined by the original function. However, it's important to note that not all functions have inverses. A function must be one-to-one (also known as injective), meaning that each output corresponds to only one input, to possess an inverse. Graphically, a one-to-one function passes the horizontal line test, where no horizontal line intersects the graph more than once.

Quadratic functions, such as the one we're examining, y = 5x² + 10, are not one-to-one over their entire domain due to the parabolic shape. This means that for a given y-value, there can be two corresponding x-values (except at the vertex). To address this, we often restrict the domain of the quadratic function to ensure it becomes one-to-one, allowing us to find a valid inverse function. This restriction typically involves considering only the positive or negative half of the parabola.

Step 1: Swapping x and y

The initial step in finding the inverse of y = 5x² + 10 is to interchange x and y. This reflects the fundamental principle of inverse functions, where the input and output roles are reversed. By swapping x and y, we set up the equation to solve for the new y, which will represent the inverse function. This gives us:

x = 5y² + 10

This equation now expresses x in terms of y, effectively setting the stage for isolating y and determining the inverse function. This swapping process is a direct application of the inverse function definition and a critical step in the overall procedure.

Step 2: Isolating the y² Term

Our next goal is to isolate the term containing y², which is 5y². This involves performing algebraic manipulations to move all other terms to the opposite side of the equation. To begin, we subtract 10 from both sides of the equation:

x - 10 = 5y²

This step isolates the term with y² on one side, bringing us closer to solving for y. Next, we divide both sides of the equation by 5 to completely isolate y²:

(x - 10) / 5 = y²

This simplifies to:

y² = (x - 10) / 5

Now, y² is isolated, paving the way for the final step of taking the square root to solve for y. This isolation is a crucial algebraic maneuver in finding the inverse.

Step 3: Solving for y

To solve for y, we need to eliminate the square. This is achieved by taking the square root of both sides of the equation:

√(y²) = ±√(( x - 10) / 5)

This gives us:

y = ±√(( x - 10) / 5)

The ± sign is crucial here because taking the square root yields both positive and negative solutions. This reflects the fact that the original quadratic function, without a restricted domain, does not have a unique inverse over its entire range. To obtain a true inverse function, we need to consider a restricted domain for the original function.

The equation y = ±√(( x - 10) / 5) represents the inverse relation. However, to express it as a function, we typically choose either the positive or negative square root, depending on the desired domain restriction of the original function. For instance, if we restricted the original function to x ≥ 0, we would choose the positive square root for the inverse function.

Step 4: Simplifying the Equation (Optional)

The equation y = ±√(( x - 10) / 5) can be further simplified, although this step is not strictly necessary for finding the inverse. Simplification often makes the equation more aesthetically pleasing and can sometimes make it easier to work with in further calculations. We can rewrite the equation as:

y = ±√( (x - 10) ) / √5

To rationalize the denominator, we multiply both the numerator and denominator by √5:

y = ±(√( (x - 10) ) * √5) / (√5 * √5)

This simplifies to:

y = ±(√(5(x - 10))) / 5

This simplified form is equivalent to the previous equation and represents the inverse relation. The choice of which form to use often depends on the specific context and the preference of the individual.

The Inverse Function and Domain Restriction

As mentioned earlier, the original quadratic function y = 5x² + 10 is not one-to-one over its entire domain. This means that to define a true inverse function, we need to restrict the domain of the original function. Let's consider two common domain restrictions:

  1. Restricting the domain to x ≥ 0: In this case, we only consider the right half of the parabola. The inverse function becomes:

    y = √(( x - 10) / 5)

    This is because the positive square root corresponds to the positive x-values in the original function.

  2. Restricting the domain to x ≤ 0: Here, we only consider the left half of the parabola. The inverse function becomes:

    y = -√(( x - 10) / 5)

    The negative square root corresponds to the negative x-values in the original function.

The domain restriction also affects the range of the inverse function and vice versa. The range of the original function becomes the domain of the inverse function, and the domain of the original function becomes the range of the inverse function. For y = 5x² + 10, the range is y ≥ 10. Therefore, the domain of the inverse function is x ≥ 10.

Conclusion

In summary, finding the inverse of the quadratic function y = 5x² + 10 involves swapping x and y, isolating y², taking the square root, and considering the domain restriction to ensure a proper inverse function is defined. The simplified equation for the inverse relation is y = ±√(( x - 10) / 5), or y = ±(√(5(x - 10))) / 5 in simplified form. By understanding the steps involved and the importance of domain restrictions, you can confidently find the inverses of various quadratic functions. Mastering this concept is a valuable asset in your mathematical journey, opening doors to more advanced topics and applications. The process of finding inverse functions is not just a mathematical exercise; it's a fundamental tool for understanding relationships between variables and solving problems in diverse fields. By grasping these concepts, you're equipping yourself with the ability to analyze and interpret the world around you through a mathematical lens.