Solving For F(-1) In The Functional Equation F(x²) + 3f(x) = 2x² - X
Introduction
In this article, we delve into the solution of a functional equation, a fascinating area of mathematics that explores the properties of functions. The specific problem at hand is to find the value of f(-1) given the functional equation f(x²) + 3f(x) = 2x² - x, where f is a function from the real numbers to the real numbers. Functional equations challenge us to think critically about the relationships between inputs and outputs of functions, and this particular equation provides an excellent example of how algebraic manipulation and substitution can lead us to a solution. Our goal is not only to find the numerical answer but also to provide a clear, step-by-step explanation of the reasoning involved, making the solution accessible and understandable. This problem showcases the beauty and elegance of mathematical problem-solving, where creativity and logical deduction intertwine to unravel the mysteries of functions.
Problem Statement
We are given the functional equation f(x²) + 3f(x) = 2x² - x, which holds for all real numbers x. Our objective is to determine the value of f(-1)*. This requires us to manipulate the given equation strategically, using suitable substitutions to isolate the desired value. The challenge lies in finding the right substitutions that will allow us to eliminate unwanted terms and eventually solve for f(-1). Let's embark on this mathematical journey and unravel the solution together.
Solution Approach
To find f(-1), we need to strategically substitute values for x in the given functional equation to create a system of equations that we can solve. The key idea here is to choose substitutions that will help us isolate f(-1). We'll start by substituting specific values that will give us equations involving f(1) and f(-1). Then, by carefully manipulating these equations, we can eliminate one of the unknowns and solve for the other. This approach demonstrates a common technique in solving functional equations: using the equation itself to generate a system of equations that can be solved using algebraic methods. This problem exemplifies how strategic substitution and algebraic manipulation can be powerful tools in the realm of functional equations.
Step 1: Substitute x = 1
First, let's substitute x = 1 into the given functional equation:
f(1²) + 3f(1) = 2(1²) - 1
Simplifying this, we get:
f(1) + 3f(1) = 2 - 1
4f(1) = 1
Therefore,
f(1) = 1/4
This substitution provides us with a crucial piece of information: the value of the function f at x = 1. This value will be instrumental in our subsequent steps as we try to find f(-1). The simplicity of this substitution belies its importance, as it lays the groundwork for the next stage of our solution. This step highlights the importance of starting with simple substitutions to gain initial insights into the behavior of the function.
Step 2: Substitute x = -1
Now, let's substitute x = -1 into the given functional equation:
f((-1)²) + 3f(-1) = 2((-1)²) - (-1)
Simplifying this, we get:
f(1) + 3f(-1) = 2 + 1
f(1) + 3f(-1) = 3
This substitution gives us another equation involving f(-1), but also includes f(1), which we already know from the previous step. This is a crucial connection that will allow us to solve for f(-1). The strategic choice of x = -1 was aimed at creating this link between the unknown f(-1) and the known f(1), demonstrating the power of targeted substitutions in solving functional equations. This step brings us closer to our goal of finding the value of f(-1).
Step 3: Substitute f(1) = 1/4
We know from Step 1 that f(1) = 1/4. Substitute this value into the equation we obtained in Step 2:
(1/4) + 3f(-1) = 3
Now we have a simple equation with only one unknown, f(-1). This is a significant milestone in our solution process, as we have successfully isolated the quantity we want to find. The substitution of the previously determined value of f(1) demonstrates how each step builds upon the previous ones, gradually leading us towards the final answer. This step showcases the importance of leveraging previously obtained results to simplify the problem.
Step 4: Solve for f(-1)
To solve for f(-1)*, we can rearrange the equation:
3f(-1) = 3 - (1/4)
3f(-1) = (12/4) - (1/4)
3f(-1) = 11/4
Now, divide both sides by 3:
f(-1) = (11/4) / 3
f(-1) = 11/12
Thus, we have found the value of f(-1)*. The algebraic manipulation in this step is straightforward but essential to isolate f(-1) and obtain the final result. The careful arithmetic ensures that we arrive at the correct answer. This step marks the culmination of our efforts, bringing us to the solution we sought.
Final Answer
The value of f(-1)* is 11/12, which is already in irreducible fraction form. Therefore, p = 11 and q = 12.
Conclusion
In this article, we successfully found the value of f(-1)* for the given functional equation f(x²) + 3f(x) = 2x² - x. We achieved this by strategically substituting values for x to create a system of equations, which we then solved using algebraic methods. The key substitutions were x = 1 and x = -1, which allowed us to find f(1) and then relate it to f(-1). This problem illustrates the power of algebraic manipulation and strategic substitution in solving functional equations. The solution process highlights the importance of breaking down a complex problem into smaller, manageable steps. Each substitution and simplification brought us closer to the final answer, demonstrating the systematic approach often required in mathematical problem-solving. This exercise not only provides a solution to a specific problem but also offers valuable insights into the techniques used to tackle functional equations in general. The result, f(-1) = 11/12, is a testament to the elegance and precision of mathematical reasoning.
