Which of the following is an even function?
[tex]f(x)=(x-1)^2[/tex]
[tex]f(x)=8 x[/tex]
[tex]f(x)=x^2-x[/tex]
[tex]f(x)=7[/tex]
Answer (2)
Check if f ( − x ) = f ( x ) for each function.
For f ( x ) = ( x − 1 ) 2 , f ( − x ) = ( x + 1 ) 2 , which is not equal to f ( x ) .
For f ( x ) = 8 x , f ( − x ) = − 8 x , which is not equal to f ( x ) .
For f ( x ) = x 2 − x , f ( − x ) = x 2 + x , which is not equal to f ( x ) .
For f ( x ) = 7 , f ( − x ) = 7 , which is equal to f ( x ) .
The even function is f ( x ) = 7 .
Explanation
Understanding Even Functions We are given four functions and we need to identify which one is an even function. A function f ( x ) is even if f ( − x ) = f ( x ) for all x in its domain. We will test each function to see if it satisfies this condition.
Testing f(x) = (x-1)^2 Let's test the first function, f ( x ) = ( x − 1 ) 2 . We need to find f ( − x ) and see if it equals f ( x ) .
f ( − x ) = ( − x − 1 ) 2 = ( − ( x + 1 ) ) 2 = ( x + 1 ) 2 = x 2 + 2 x + 1
f ( x ) = ( x − 1 ) 2 = x 2 − 2 x + 1
Since x 2 + 2 x + 1 e q x 2 − 2 x + 1 , the function f ( x ) = ( x − 1 ) 2 is not even.
Testing f(x) = 8x Now let's test the second function, f ( x ) = 8 x .
f ( − x ) = 8 ( − x ) = − 8 x
Since − 8 x e q 8 x (unless x = 0 , but the condition must hold for all x ), the function f ( x ) = 8 x is not even.
Testing f(x) = x^2 - x Next, let's test the third function, f ( x ) = x 2 − x .
f ( − x ) = ( − x ) 2 − ( − x ) = x 2 + x
Since x 2 + x e q x 2 − x (unless x = 0 , but the condition must hold for all x ), the function f ( x ) = x 2 − x is not even.
Testing f(x) = 7 Finally, let's test the fourth function, f ( x ) = 7 .
f ( − x ) = 7
Since 7 = 7 , the function f ( x ) = 7 is an even function.
Conclusion Therefore, the even function among the given options is f ( x ) = 7 .
Examples
Even functions are symmetric about the y-axis, meaning their graph looks the same on both sides of the y-axis. This property is useful in physics and engineering, where symmetry simplifies calculations and helps in understanding the behavior of systems. For example, in signal processing, even functions are used to represent signals that have time-reversal symmetry, which can simplify the analysis of these signals. The constant function f ( x ) = 7 is a simple example of an even function, and its symmetry makes it easy to visualize and understand.
The even function among the given options is f ( x ) = 7 because it satisfies the condition f ( − x ) = f ( x ) for all x . The other functions do not satisfy this condition. Thus, the answer is f ( x ) = 7 .
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