An even function is a mathematical function that satisfies the property f(x) = f(-x) for all x in its domain. In simpler terms, an even function is symmetric with respect to the y-axis. This article will discuss what even functions are and provide examples of even functions.
Characteristics of an Even Function
An even function has the following characteristics:
- Symmetry: An even function is symmetric with respect to the y-axis. This means that if you reflect the graph of the function across the y-axis, the resulting graph will be identical.
- Property: The property of an even function is f(x) = f(-x) for all x in its domain. This property defines the symmetric nature of even functions.
- Examples: Examples of even functions include cosine function, exponential function, and parabola with its vertex on the y-axis.
Examples of Even Functions
Now, let’s look at some examples of even functions:
- Cosine Function: The cosine function, denoted as cos(x), is an even function. It is symmetric with respect to the y-axis, and cos(x) = cos(-x) for all x.
- Exponential Function: The exponential function, denoted as exp(x) or e^x, is also an even function. It is symmetric with respect to the y-axis, and exp(x) = exp(-x) for all x.
- Parabola: A parabola with its vertex on the y-axis is an even function. The function f(x) = x^2 is an example of an even function.
Testing for Even Functions
To determine if a function is even, you can apply the following steps:
- Substitute -x for x: Replace x with -x in the function and simplify the expression.
- Compare f(x) and f(-x): Check if the resulting expressions are equal. If f(x) = f(-x), then the function is even.
Which Of The Following Is An Even Function?
Consider the following functions and determine which of them are even functions:
- f(x) = x^4 – 3x^2 + 2:
- g(x) = 2sin(x):
To test if this function is even, we substitute -x for x:
f(-x) = (-x)^4 – 3(-x)^2 + 2
f(-x) = x^4 – 3x^2 + 2
Since f(x) = f(-x), the function f(x) = x^4 – 3x^2 + 2 is an even function.
To test if this function is even, we substitute -x for x:
g(-x) = 2sin(-x)
g(-x) = -2sin(x)
The function g(x) = 2sin(x) is not even, as g(x) is not equal to g(-x).
Conclusion
In conclusion, even functions are mathematical functions that exhibit symmetry with respect to the y-axis. They satisfy the property f(x) = f(-x) for all x in their domain. Examples of even functions include cosine function, exponential function, and parabolas with vertices on the y-axis. Testing for even functions involves substituting -x for x and comparing the resulting expressions. By understanding the characteristics and examples of even functions, you can easily identify them in various mathematical contexts.
