Which of the following is an even function?

[tex]g(x)=(x-1)^2+1[/tex]
[tex]g(x)=2 x^2+1[/tex]
[tex]g(x)=4 x+2[/tex]
[tex]g(x)=2 x[/tex]

An even function satisfies the condition g ( x ) = g ( − x ) .
Test each function by substituting − x for x .
For g ( x ) = ( x − 1 ) 2 + 1 , g ( − x ) = x 2 + 2 x + 2 , which is not equal to g ( x ) .
For g ( x ) = 2 x 2 + 1 , g ( − x ) = 2 x 2 + 1 , which is equal to g ( x ) . Thus, g ( x ) = 2 x 2 + 1 is an even function.
For g ( x ) = 4 x + 2 , g ( − x ) = − 4 x + 2 , which is not equal to g ( x ) .
For g ( x ) = 2 x , g ( − x ) = − 2 x , which is not equal to g ( x ) .
The even function is 2 x 2 + 1 ​ .

Explanation

Understanding Even Functions We are given four functions and we need to identify which one is even. A function is even if replacing x with − x results in the same function, i.e., g ( x ) = g ( − x ) . Let's test each function.

Testing the first function

g ( x ) = ( x − 1 ) 2 + 1 Let's find g ( − x ) : g ( − x ) = ( − x − 1 ) 2 + 1 = ( − ( x + 1 ) ) 2 + 1 = ( x + 1 ) 2 + 1 = x 2 + 2 x + 1 + 1 = x 2 + 2 x + 2 Since g ( x ) = ( x − 1 ) 2 + 1 = x 2 − 2 x + 1 + 1 = x 2 − 2 x + 2 , we see that g ( − x ) e q g ( x ) . So, this function is not even.

Testing the second function

g ( x ) = 2 x 2 + 1 Let's find g ( − x ) : g ( − x ) = 2 ( − x ) 2 + 1 = 2 ( x 2 ) + 1 = 2 x 2 + 1 Since g ( − x ) = g ( x ) , this function is even.

Testing the third function

g ( x ) = 4 x + 2 Let's find g ( − x ) : g ( − x ) = 4 ( − x ) + 2 = − 4 x + 2 Since g ( − x ) e q g ( x ) , this function is not even.

Testing the fourth function

g ( x ) = 2 x Let's find g ( − x ) : g ( − x ) = 2 ( − x ) = − 2 x Since g ( − x ) e q g ( x ) , this function is not even.

Conclusion Therefore, the only even function among the given options is g ( x ) = 2 x 2 + 1 .


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 phenomena. For example, in signal processing, even functions represent signals that are symmetrical in time, which can simplify analysis and filtering processes. Similarly, in structural mechanics, symmetrical loads on symmetrical structures result in even functions describing stress distribution, making the analysis easier.

Answered by GinnyAnswer | 2025-07-03

The even function among the given options is g ( x ) = 2 x 2 + 1 , as it satisfies the condition g ( x ) = g ( − x ) . The other functions do not meet this criteria. Therefore, the answer is 2 x 2 + 1 .
;

Answered by Anonymous | 2025-07-04