Even Function Derived From Odd Function G(x) Exploration
In the realm of mathematical functions, the concepts of odd and even functions hold significant importance. Understanding their properties and how they interact is crucial for solving various problems. This article delves into the characteristics of odd and even functions and explores how to identify an even function when given an odd function, g(x). We will analyze the given options to determine which one must result in an even function.
Understanding Odd and Even Functions
Before we dive into the problem, let's establish a clear understanding of what constitutes odd and even functions. This foundational knowledge is essential for correctly identifying the even function derived from g(x). Key properties define these functions, and grasping them is vital for success in this problem and similar mathematical challenges.
Even Functions
An even function is defined by its symmetry about the y-axis. Mathematically, this means that for any value of x, the function value at x is the same as the function value at -x. In other words, if you were to fold the graph of the function along the y-axis, the two halves would perfectly overlap. The mathematical representation of an even function is:
f(x) = f(-x)
A classic example of an even function is f(x) = x². If you substitute x with -x, you get f(-x) = (-x)² = x², which is the same as f(x). Other examples include cos(x) and any polynomial function containing only even powers of x.
The symmetry of even functions simplifies many mathematical operations, particularly in calculus and Fourier analysis. Their graphs are mirror images across the y-axis, making them visually and conceptually straightforward. Recognizing even functions can significantly streamline problem-solving in various mathematical contexts.
Odd Functions
An odd function, in contrast to an even function, exhibits symmetry about the origin. This means that for any value of x, the function value at x is the negative of the function value at -x. Visually, this implies that if you rotate the graph of the function 180 degrees about the origin, it remains unchanged. The mathematical representation of an odd function is:
f(-x) = -f(x)
A common example of an odd function is f(x) = x. Substituting x with -x gives f(-x) = -x, which is the negative of f(x). Other examples include sin(x) and any polynomial function containing only odd powers of x. The graph of an odd function has rotational symmetry around the origin, meaning it looks the same after a 180-degree rotation.
Understanding the symmetry of odd functions is crucial in various mathematical fields, including calculus and differential equations. The properties of odd functions often lead to simplifications in complex calculations, making them an essential concept in mathematical analysis.
Analyzing the Options
Now that we have a solid understanding of odd and even functions, let's analyze each option provided in the problem to determine which one must be an even function when g(x) is an odd function. This involves applying the definitions and properties of odd and even functions to each case and carefully evaluating the results.
Option A: f(x) = g(x) + 2
Let's examine f(-x) for this function:
f(-x) = g(-x) + 2
Since g(x) is an odd function, we know that g(-x) = -g(x). Substituting this into the equation, we get:
f(-x) = -g(x) + 2
Now, we compare f(-x) with f(x):
f(x) = g(x) + 2 f(-x) = -g(x) + 2
Clearly, f(x) is not equal to f(-x), so this function is not even. Additionally, f(-x) is not equal to -f(x), so it is also not odd. Therefore, adding a constant to an odd function generally results in a function that is neither even nor odd.
Option A does not yield an even function.
Option B: f(x) = g(x) + g(x)
This can be simplified to f(x) = 2g(x). Now, let's find f(-x):
f(-x) = 2g(-x)
Since g(x) is an odd function, g(-x) = -g(x). Substituting this, we get:
f(-x) = 2[-g(x)] = -2g(x)
Comparing f(-x) with f(x):
f(x) = 2g(x) f(-x) = -2g(x)
Here, f(-x) = -f(x), which means f(x) is an odd function, not an even function.
Option B also does not result in an even function.
Option C: f(x) = g(x)²
Let's find f(-x):
f(-x) = [g(-x)]²
Since g(x) is an odd function, g(-x) = -g(x). Substituting this, we get:
f(-x) = [-g(x)]² = g(x)²
Comparing f(-x) with f(x):
f(x) = g(x)² f(-x) = g(x)²
Here, f(x) = f(-x), which perfectly matches the definition of an even function. Squaring an odd function always results in an even function because the negative sign is eliminated during the squaring process. This is a fundamental property that makes Option C the correct choice.
Option C results in an even function.
Option D: f(x) = -g(x)
To determine if this function is even, let's evaluate f(-x):
f(-x) = -g(-x)
Since g(x) is an odd function, we know that g(-x) = -g(x). Substituting this into the equation, we get:
f(-x) = -[-g(x)] = g(x)
Now, comparing f(-x) with f(x):
f(x) = -g(x) f(-x) = g(x)
We can see that f(-x) = -f(x), which means f(x) is an odd function, not an even function. Multiplying an odd function by -1 simply reflects the function across the x-axis, maintaining its odd symmetry.
Option D does not yield an even function.
Conclusion
After thoroughly analyzing each option, we can confidently conclude that if g(x) is an odd function, the function that must be an even function is Option C: f(x) = g(x)². This is because squaring an odd function eliminates the negative sign associated with the odd function's symmetry about the origin, resulting in a function that is symmetric about the y-axis, which is the defining characteristic of an even function. This exploration underscores the importance of understanding the fundamental properties of odd and even functions and how they interact under different mathematical operations.
Final Answer
Therefore, the final answer is C. f(x) = g(x)².