Inverse Function Value: Calculation Examples

Jan 14, 2026 by Tim Redaksi 45 views

Introduction

In mathematics, especially in calculus and algebra, inverse functions play a crucial role. Understanding inverse functions is essential for solving various problems, including finding specific values. Guys, in this article, we're going to break down how to calculate the value of inverse functions given certain conditions. We'll tackle two examples step by step, so you can follow along and get a solid grasp of the concept. Whether you're a student prepping for an exam or just someone looking to brush up on your math skills, this guide is for you. Let's dive in and make inverse functions less intimidating together!

Problem 1: Finding g⁻¹(0) when g(x) = (x + 2)/(3x + 1)

Understanding the Problem

The first problem asks us to find the value of the inverse function g⁻¹(0) given the function g(x) = (x + 2)/(3x + 1), where x ≠ -1/3. This means we need to find the value of x for which g(x) equals 0. In other words, we are solving the equation (x + 2)/(3x + 1) = 0. This requires us to understand how to manipulate and solve rational equations. The restriction x ≠ -1/3 is important because it ensures that the denominator of the original function is never zero, thus avoiding undefined values. This kind of problem often appears in algebra and calculus courses, testing your ability to work with functions and their inverses. So, let’s get started and solve it step by step!

Step-by-Step Solution

Set up the equation: We start by setting g(x) equal to 0: (x + 2)/(3x + 1) = 0
Solve for x: To solve this equation, we need to find the value of x that makes the fraction equal to zero. A fraction is zero only when its numerator is zero. Therefore, we set the numerator equal to zero: x + 2 = 0 Subtract 2 from both sides: x = -2
Check the solution: We need to make sure that x = -2 does not make the denominator 3x + 1 equal to zero. If it did, the function would be undefined at that point. 3x + 1 = 3(-2) + 1 = -6 + 1 = -5 Since the denominator is not zero, x = -2 is a valid solution.
Find the inverse function value: Since g(-2) = 0, it follows that g⁻¹(0) = -2.

Conclusion for Problem 1

Therefore, the value of g⁻¹(0) for the given function g(x) = (x + 2)/(3x + 1) is -2. This means that when the output of the inverse function is 0, the input is -2. Understanding these steps helps in solving similar problems involving rational functions and their inverses. It's all about setting up the equation correctly and solving for the unknown variable. Remember, always check your solution to make sure it doesn't make the original function undefined. Great job, guys! Let's move on to the next problem.

Problem 2: Finding g⁻¹(2) when g(x) = (1/2)x

Understanding the Problem

In this problem, we are asked to find the value of the inverse function g⁻¹(2) given the function g(x) = (1/2)x, where x ≠ 0. This means we need to find the value of x for which g(x) equals 2. In other words, we are solving the equation (1/2)x = 2. This is a simpler linear equation compared to the first problem, but it still requires a clear understanding of inverse functions. The restriction x ≠ 0 is not particularly relevant in this case, as it only affects the original function's definition, not the solution to our problem. This kind of problem is common in introductory algebra courses, focusing on linear functions and their properties. Let's break it down step by step!

Step-by-Step Solution

Set up the equation: We start by setting g(x) equal to 2: (1/2)x = 2
Solve for x: To solve for x, we need to isolate x on one side of the equation. We can do this by multiplying both sides of the equation by 2: 2 * (1/2)x = 2 * 2 x = 4
Check the solution: We need to make sure that x = 4 does not violate any restrictions. Since there are no restrictions that x cannot be 4, this solution is valid.
Find the inverse function value: Since g(4) = 2, it follows that g⁻¹(2) = 4.

Conclusion for Problem 2

Therefore, the value of g⁻¹(2) for the given function g(x) = (1/2)x is 4. This means that when the output of the inverse function is 2, the input is 4. Understanding these steps helps in solving similar problems involving linear functions and their inverses. It's all about setting up the equation correctly and solving for the unknown variable. Remember, always check your solution to make sure it doesn't violate any restrictions. Great job, guys! You've tackled another problem successfully.

General Conclusion

In conclusion, finding the value of an inverse function involves setting the original function equal to the desired output value and then solving for the input variable. Whether dealing with rational functions or linear functions, the process remains the same: set up the equation, solve for x, and check your solution. These examples illustrate the fundamental principles of inverse functions and their applications. Keep practicing, and you'll become more confident in solving these types of problems. Remember, math is all about practice, practice, practice! So keep at it, guys, and you'll ace those exams and impress your friends with your math skills. Understanding inverse functions opens the door to more advanced topics in mathematics, so it's definitely worth the effort. Keep up the great work!