Solving the Equation ( f(x) f(-x) ) and Understanding Its Implications

When we encounter the equation ( f(x) f(-x) ), we are dealing with a very specific type of function, known as an even function. Let's explore the process of solving this equation and the implications of its solutions.

Solving for ( f(x) f(-x) )

First, let's break down the equation ( f(x) f(-x) ).

Given the equation: [ f(x) f(-x) ] Let's define ( g(x) ) as an arbitrary function, and express ( f(x) ) in terms of ( g(x) ). The general form of the solution is: [ f(x) frac{g(x) g(-x)}{2} ] This means that ( g(x) g(-x) ) must be an even function, and dividing it by 2 results in another even function.

To solve the equation, let's use a step-by-step approach:

Step 1: Set up the equation. [ f(x) f(-x) ] Step 2: Assume ( g(x) ) is an arbitrary function. Then, we can rewrite the equation as: [ g(x) g(-x) ] Step 3: From the properties of ( g(x) ), we know that any ( g(x) ) that satisfies ( g(x) g(-x) ) represents an even function. Therefore, we can express ( f(x) ) as: [ f(x) frac{g(x) g(-x)}{2} g(x) ] Step 4: Check the solution by substituting ( x 0 ) into the original equation. [ f(0) f(-0) ] Since ( -0 0 ), the equation holds true, confirming our solution.

The solution to the equation ( f(x) f(-x) ) is an even function, meaning ( f(x) ) is symmetric with respect to the y-axis.

Examples of Even Functions

Let's explore some specific examples of even functions:

Example 1: ( f(x) cos x ) [ cos( -x) cos(x) ] Example 2: ( f(x) |x| ) (absolute value function) [ | -x | |x| ] Example 3: ( f(x) 2x^2 ) (vertical parabola) [ 2(-x)^2 2x^2 ] Example 4: ( f(x) frac{x^2}{3} ) [ frac{(-x)^2}{3} frac{x^2}{3} ] Example 5: ( f(x) frac{1}{x^2} ) [ frac{1}{(-x)^2} frac{1}{x^2} ]

Note that each of these examples satisfies the property ( f(x) f(-x) ), confirming that they are even functions.

Graphical Interpretation of Even Functions

From a graphical perspective, every even function has a symmetrical graph with respect to the y-axis. If ( M(x, y) ) is a point on the graph of ( f(x) ), then the point ( M(-x, y) ) is also on the graph, as shown by the following property:

For any even function ( f(x) ), if ( M(x, y) in G_f ), then ( M(-x, y) in G_f ).

For instance, consider the graph of ( f(x) cos x ) and the graph of ( f(x) frac{1}{x^2} ). Both graphs are symmetric with respect to the y-axis, as demonstrated below.

Graph of even functions showing symmetry about the y-axis

Factor to Consider

While the equation ( f(x) f(-x) ) is not a traditional equation that needs to be solved for a variable, it serves as a definition for even functions. The key point here is that any function that satisfies this condition is symmetric with respect to the y-axis. This property is crucial in various mathematical fields, such as calculus and mathematical analysis.

For more detailed information, you can refer to textbooks on calculus or mathematical analysis, which often cover the topic of even functions in the chapter on elementary functions.

Conclusion

The equation ( f(x) f(-x) ) represents the definition of an even function. By exploring this equation, we gain a deeper understanding of symmetry in functions and their graphical representations. Understanding even functions is essential in fields such as calculus, where symmetry can simplify many problems.