Which of the following is an even function?
$f(x)=(x-1)^2$
$f(x)=8 x$
$f(x)=x^2-x$
$f(x)=7$
Answer (1)
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 7 .
Explanation
Understanding the problem We are given four functions and we need to determine which one is even. A function is even if f ( − x ) = f ( x ) for all x . We will check each function.
Checking the first function Let's analyze the first function: f ( x ) = ( x − 1 ) 2 . We need to find f ( − x ) .
f ( − x ) = ( − x − 1 ) 2 = ( − ( x + 1 ) ) 2 = ( x + 1 ) 2 = x 2 + 2 x + 1 . Now we check if f ( − x ) = f ( x ) .
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.
Checking the second function Now let's analyze the second function: f ( x ) = 8 x . We need to find f ( − x ) .
f ( − x ) = 8 ( − x ) = − 8 x . Now we check if f ( − x ) = f ( x ) .
Since − 8 x e q 8 x , the function f ( x ) = 8 x is not even.
Checking the third function Now let's analyze the third function: f ( x ) = x 2 − x . We need to find f ( − x ) .
f ( − x ) = ( − x ) 2 − ( − x ) = x 2 + x . Now we check if f ( − x ) = f ( x ) .
Since x 2 + x e q x 2 − x , the function f ( x ) = x 2 − x is not even.
Checking the fourth function Now let's analyze the fourth function: f ( x ) = 7 . We need to find f ( − x ) .
f ( − x ) = 7 . Now we check if f ( − x ) = f ( x ) .
Since 7 = 7 , the function f ( x ) = 7 is even.
Conclusion Therefore, the even function 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. In physics, potential energy functions are often even functions, simplifying calculations. For example, the potential energy of a simple harmonic oscillator is proportional to x 2 , which is an even function. Understanding even functions helps in simplifying complex problems in various fields.
