Finding The Inverse Of Y=7x^2-10 A Step-by-Step Guide

In mathematics, the concept of an inverse function is crucial for understanding the relationship between different equations and their transformations. The inverse of a function essentially reverses the operation performed by the original function. When we talk about the inverse of an equation like y = 7x² - 10, we are looking for a new equation that expresses x in terms of y. This process involves swapping the roles of x and y and then solving for y. Understanding how to find the inverse of a function is a fundamental skill in algebra and calculus, with applications spanning various fields, including physics, engineering, and computer science. This article will delve into the step-by-step method of finding the inverse of the given quadratic function and clarify the nuances involved in this mathematical operation.

Before diving into the specifics of the given equation, let's first establish a clear understanding of what inverse functions are and how they operate. Inverse functions are essentially the 'undoing' of the original function. Mathematically, if we have a function f(x), its inverse, denoted as f⁻¹(x), satisfies the condition that f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. In simpler terms, if you apply a function and then apply its inverse, you should end up with the original input. For a function to have a true inverse, it must be bijective, meaning it is both injective (one-to-one) and surjective (onto). However, when dealing with functions like the quadratic equation in question, which are not one-to-one across their entire domain, we often restrict the domain to create a section where the inverse can be properly defined. The process of finding an inverse involves several algebraic manipulations, primarily swapping the variables x and y and then isolating y. This transformation gives us a new equation that represents the inverse function, allowing us to understand the reverse mapping of the original function.

To find the inverse of the equation y = 7x² - 10, we follow a systematic approach that involves several algebraic steps. Let's break down the process:

1. Swap x and y: The first step in finding the inverse is to interchange the variables x and y. This reflects the fundamental concept of an inverse function, which reverses the roles of input and output. So, we replace y with x and x with y in the original equation. This gives us x = 7y² - 10.

2. Isolate the y² term: Next, we need to isolate the term containing y² to make it easier to solve for y. To do this, we add 10 to both sides of the equation: x + 10 = 7y². This step moves the constant term to the side with x, bringing us closer to isolating y².

3. Divide by the coefficient of y²: Now, we divide both sides of the equation by 7 to isolate y² completely. This gives us (x + 10) / 7 = y². This step is crucial in simplifying the equation and preparing it for the final step of solving for y.

4. Take the square root: To solve for y, we take the square root of both sides of the equation. When taking the square root, it’s essential to remember that we need to consider both the positive and negative roots, as both will satisfy the equation. This yields y = ±√((x + 10) / 7). The inclusion of both positive and negative roots is a key aspect of finding the inverse of a quadratic function, as it reflects the symmetry of the parabola.

Therefore, the inverse of the function y = 7x² - 10 is y = ±√((x + 10) / 7).

Now that we have found the inverse function, let's compare our result with the given options to identify the correct answer. The options are:

A. y = (±√(x + 10)) / 7 B. y = ±√((x + 10) / 7) C. y = ±√(x/7 + 10) D. y = (±√x) / 7 ± (√10) / 7

Comparing our derived inverse function, y = ±√((x + 10) / 7), with the options, we can clearly see that:

• Option A is incorrect because it places the division by 7 outside the square root.
• Option B matches our result exactly, making it the correct inverse function.
• Option C is incorrect as it distributes the division incorrectly inside the square root.
• Option D is also incorrect as it attempts to separate the terms inside the square root in an invalid manner.

Therefore, the correct option is B: y = ±√((x + 10) / 7). This meticulous comparison ensures that we have correctly identified the inverse function by matching our solution with the provided choices.

Graphically, understanding the inverse function provides another layer of insight into the relationship between the original function and its inverse. The graph of a function and its inverse are reflections of each other across the line y = x. This means that if you were to draw the line y = x on the same graph as y = 7x² - 10 and its inverse y = ±√((x + 10) / 7), the two curves would appear as mirror images across this line. Visualizing this reflection helps to confirm the correctness of the inverse function we derived algebraically. The original quadratic function y = 7x² - 10 is a parabola opening upwards with its vertex at (0, -10). Its inverse, however, comprises two parts due to the ± sign, representing the reflection of the parabola about the line y = x. The positive square root part represents the upper half of the reflected parabola, while the negative square root part represents the lower half. This graphical representation not only validates our algebraic solution but also enhances our understanding of how inverse functions transform and relate to their original functions.

When discussing inverse functions, it’s essential to consider the domain and range of both the original function and its inverse. The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values). For the original function y = 7x² - 10, the domain is all real numbers because we can input any real number for x. However, the range is y ≥ -10 because the square of any real number is non-negative, and thus, the minimum value of y is -10. When we find the inverse y = ±√((x + 10) / 7), the domain and range are swapped. The domain of the inverse function is x ≥ -10, which is the range of the original function, and the range of the inverse function is all real numbers, which is the domain of the original function. However, due to the ± sign in the inverse function, we must consider that this represents two separate functions, each with a restricted range to ensure they are true inverses in a bijective sense. Specifically, we often consider y = √((x + 10) / 7) and y = -√((x + 10) / 7) separately, each having a specific domain and range that corresponds to a portion of the original function's domain and range. This detailed consideration of domain and range is crucial for a complete understanding of inverse functions and their properties.

Finding the inverse of a function can sometimes be tricky, and there are several common mistakes that students often make. Recognizing and avoiding these pitfalls is crucial for ensuring accurate results. One of the most frequent errors is forgetting to consider both the positive and negative square roots when solving for y. In the case of y = 7x² - 10, the inverse involves taking a square root, and it’s imperative to include both ± to account for the two possible solutions. Another common mistake is incorrectly performing algebraic manipulations, such as distributing a division inside a square root or incorrectly isolating terms. For example, √(a + b) is not equal to √a + √b, and failing to recognize this can lead to errors. Additionally, students sometimes forget to swap x and y at the beginning of the process, which is a fundamental step in finding the inverse. Another subtle mistake is not considering the domain and range restrictions. While algebraically an inverse might be found, it may not be a true inverse over the entire set of real numbers due to the function not being one-to-one. Therefore, understanding the domain and range and potentially restricting them is essential for a complete and accurate solution. By being mindful of these common mistakes, you can improve your accuracy and confidence in finding inverse functions.

In conclusion, finding the inverse of the equation y = 7x² - 10 involves a methodical process of swapping variables, isolating y, and considering both positive and negative roots. Through our step-by-step solution, we determined that the correct inverse function is y = ±√((x + 10) / 7), which corresponds to option B. Understanding the graphical representation, domain and range considerations, and common mistakes to avoid further solidifies the concept of inverse functions. The inverse function reverses the operation of the original function, and this understanding is a cornerstone of many mathematical concepts. Mastering the process of finding inverses enhances your problem-solving skills and provides a deeper appreciation of the relationships between different mathematical expressions. Whether for academic pursuits or practical applications, the ability to find and interpret inverse functions is a valuable asset in mathematics and beyond.