Common Mistakes to Avoid When Identifying Even Functions

Oct 13, 2025 by miodeya.com 95 views

Introduction

In mathematics, an even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. This means that if a function is even, then its graph is symmetric with respect to the y-axis. Even functions have many important applications in various fields, including physics, engineering, and computer science. In this article, we will discuss which of the given functions is an even function.

What is an Even Function?

An even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. This means that if a function is even, then its graph is symmetric with respect to the y-axis. To determine if a function is even, we can use the following criteria:

If f(-x) = f(x) for all x in the domain of the function, then the function is even.
If the graph of the function is symmetric with respect to the y-axis, then the function is even.

Analyzing the Options

Now, let's analyze the given options to determine which one is an even function.

A. $f(x) = (x-1)^2$

To determine if this function is even, we need to substitute -x for x and simplify the expression.

$f(-x) = ((-x)-1)^2$ $f(-x) = (-x-1)^2$ $f(-x) = (x+1)^2$

Since $f(-x) \neq f(x)$ , this function is not even.

B. $f(x) = 8x$

To determine if this function is even, we need to substitute -x for x and simplify the expression.

$f(-x) = 8(-x)$ $f(-x) = -8x$

Since $f(-x) \neq f(x)$ , this function is not even.

C. $f(x) = x^2 - x$

To determine if this function is even, we need to substitute -x for x and simplify the expression.

$f(-x) = (-x)^2 - (-x)$ $f(-x) = x^2 + x$

Since $f(-x) \neq f(x)$ , this function is not even.

D. $f(x) = 7$

To determine if this function is even, we need to substitute -x for x and simplify the expression.

$f(-x) = 7$

Since $f(-x) = f(x)$ , this function is even.

Conclusion

In conclusion, the only even function among the given options is $f(x) = 7$ . This function satisfies the condition f(-x) = f(x) for all x in its domain, and its graph is symmetric with respect to the y-axis.

Why is it Important to Identify Even Functions?

Identifying even functions is important in many areas of mathematics and science. For example, even functions are used to model physical systems that are symmetric with respect to the y-axis, such as the motion of a pendulum. Even functions are also used in signal processing and image analysis, where they are used to filter out noise and enhance the signal.

Real-World Applications of Even Functions

Even functions have many real-world applications, including:

Physics: Even functions are used to model the motion of objects that are symmetric with respect to the y-axis, such as the motion of a pendulum.
Engineering: Even functions are used in signal processing and image analysis to filter out noise and enhance the signal.
Computer Science: Even functions are used in algorithms for image and signal processing, such as the Fast Fourier Transform (FFT) algorithm.

Common Mistakes to Avoid When Identifying Even Functions

When identifying even functions, there are several common mistakes to avoid:

Not checking the condition f(-x) = f(x) for all x in the domain: This is the most common mistake when identifying even functions. Make sure to check the condition for all x in the domain.
Not considering the symmetry of the graph: Even functions have graphs that are symmetric with respect to the y-axis. Make sure to consider the symmetry of the graph when identifying even functions.

Conclusion

In conclusion, identifying even functions is an important concept in mathematics and science. By understanding what an even function is and how to identify one, we can apply this knowledge to many real-world problems. Remember to check the condition f(-x) = f(x) for all x in the domain and consider the symmetry of the graph when identifying even functions.