Solving Equations & Proving Even Functions: A Math Guide

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Solving Equations & Proving Even Functions: A Math Guide

Hey math enthusiasts! Let's dive into some cool problems involving functions and equations. We'll be tackling how to find specific values of x that yield the same function output, and then we'll learn to identify and prove whether a function is even. Buckle up, it's going to be a fun ride!

Understanding the Core Concepts

Before we jump into the problems, let's make sure we're all on the same page. This section serves as a refresher for basic concepts like quadratic functions and the definition of even functions. You know, the fundamentals!

Quadratic functions, which are functions whose formula can be written as ax² + bx + c = 0 where a, b and c are real numbers, and a is not equal to zero. This form is essential because it helps to understand the behaviour of the function. For example, the function creates a parabola. The function f(x) = a(x - 2)² + 41 is basically a quadratic function, but written in vertex form, which is super convenient for finding the vertex of the parabola.

Now, about even functions. A function is considered even if it satisfies a specific property: f(-x) = f(x) for all values of x in its domain. Basically, if you plug in x or -x into the function, you get the same result. The graph of an even function is symmetric with respect to the y-axis. It's like looking in a mirror where the y-axis is the mirror. If you fold the graph along the y-axis, the two sides will perfectly overlap. Understanding these concepts is the key to solving the problems we're about to face, so make sure you've got them down. It helps tremendously!

In our problems, we will use these concepts to determine the values of x and also to find out if the provided functions are even or not. So, keep an eye on these concepts as you go through the examples. Remembering them can make complex problems a lot more manageable.

Finding x1 and x2 When f(x1) = f(x2)

Let's kick things off by figuring out how to find the values of x that make the function outputs equal. We're essentially solving equations. We'll start with the function f(x) = a(x - 2)² + 41. The question here is: find real numbers x1 and x2 such that f(x1) = f(x2). Note that to do this, we need to know the value of 'a'.

For the first function we have f(x) = x² - 2x + 7. To find x1 and x2, you might have to first realize that this is a quadratic function and work from there. The trick is to set up the equation like this: since we need f(x1) = f(x2), and because f(x) = x² - 2x + 7, let's assume x1 and x2 are distinct values. That would mean (x1)² - 2x1 + 7 = (x2)² - 2x2 + 7. Simplifying, we get (x1)² - 2x1 = (x2)² - 2x2. Then, (x1)² - (x2)² - 2x1 + 2x2 = 0. This can be factored into (x1 - x2)(x1 + x2) - 2(x1 - x2) = 0. Then, (x1 - x2)(x1 + x2 - 2) = 0. Now, this equation tells us that either x1 - x2 = 0 or x1 + x2 - 2 = 0. The first case indicates x1 = x2, which would not yield two distinct numbers. The second one implies x1 + x2 = 2. So, finding x1 and x2 would mean solving for the same equation with two unknowns. For example, if x1 is 0, x2 would be 2. Or if x1 is 1, x2 would be 1. It is good to remember that in this case, we have a function and it is not required that we provide specific values, it's just to express a relationship between x1 and x2.

Now, for the last function, f(x) = x² + 4x - 3. We follow a similar approach as before. Again, if f(x1) = f(x2), then (x1)² + 4x1 - 3 = (x2)² + 4x2 - 3. Simplifying, we get (x1)² + 4x1 = (x2)² + 4x2. This can be rearranged into (x1)² - (x2)² + 4x1 - 4x2 = 0, which can be factored to (x1 - x2)(x1 + x2) + 4(x1 - x2) = 0. Then, (x1 - x2)(x1 + x2 + 4) = 0. As we did before, either x1 - x2 = 0, which means x1 = x2, or x1 + x2 + 4 = 0. Which means x1 + x2 = -4. This demonstrates the relationship between x1 and x2. Again, we have a function and it is not required that we provide specific values, it's just to express a relationship between x1 and x2.

Proving if Functions are Even

Now, let's dive into proving whether the functions are even. Remember, a function is even if f(-x) = f(x) for all x. We have to demonstrate that the output of the function is the same when we use x or -x. Let's break it down function by function.

For the function f(x) = x² - 2x + 7, we need to see if f(-x) = f(x). Let's substitute -x into the function: f(-x) = (-x)² - 2(-x) + 7 = x² + 2x + 7. Now, let's compare f(-x) with f(x). We can see that x² + 2x + 7 is not equal to x² - 2x + 7. The function does not have symmetry with respect to the y-axis, and therefore it is not an even function. Therefore, because f(-x) does not equal f(x), the function f(x) = x² - 2x + 7 is not an even function.

For the function f(x) = x² + 4x - 3, again, we need to check if f(-x) = f(x). Substitute -x into the function: f(-x) = (-x)² + 4(-x) - 3 = x² - 4x - 3. Let's compare f(-x) with f(x). We see that x² - 4x - 3 is not the same as x² + 4x - 3. Hence, the function f(x) = x² + 4x - 3 is not an even function. Proving that a function is not even is usually done by demonstrating that the condition f(-x) = f(x) does not hold. This is a crucial step in understanding the function's behaviour.

Summary and Key Takeaways

So, there you have it! We've tackled the problem of finding x1 and x2 when function outputs are equal, and we've learned how to test and prove whether a function is even. Here's a quick recap of the key takeaways:

  • Understanding the Definitions: Make sure you know what a quadratic function is and the properties of an even function.
  • Setting up Equations: To find x1 and x2, you often need to set up and solve equations based on the function's formula.
  • Testing for Even Functions: To prove a function is even, verify that f(-x) = f(x). If it doesn't, the function is not even.
  • Practice is Key: The more problems you solve, the more comfortable you'll become with these concepts.

Keep practicing, and you'll become a pro at these math problems in no time. If you have any questions, feel free to ask! Happy solving, guys!