Which functions are even? Check all of the boxes that apply.
[tex]f(x)=x^4-x^2[/tex]
[tex]f(x)=x^2-3 x+2[/tex]
[tex]f(x)=\sqrt{(x-2)}[/tex]
[tex]f(x)=|x|[/tex]
Answer (2)
The even functions are f ( x ) = x 4 − x 2 and f ( x ) = ∣ x ∣ . The function f ( x ) = x 2 − 3 x + 2 is not even, and neither is f ( x ) = x − 2 due to domain issues. Therefore, the chosen options are the first and last functions listed.
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Check if f ( − x ) = f ( x ) for each function.
For f ( x ) = x 4 − x 2 , f ( − x ) = ( − x ) 4 − ( − x ) 2 = x 4 − x 2 = f ( x ) , so it's even.
For f ( x ) = x 2 − 3 x + 2 , f ( − x ) = ( − x ) 2 − 3 ( − x ) + 2 = x 2 + 3 x + 2 = f ( x ) , so it's not even.
For f ( x ) = x − 2 , the domain of f ( x ) is x ≥ 2 and the domain of f ( − x ) is x ≤ − 2 , so it's not even.
For f ( x ) = ∣ x ∣ , f ( − x ) = ∣ − x ∣ = ∣ x ∣ = f ( x ) , so it's even.
The even functions are f ( x ) = x 4 − x 2 and f ( x ) = ∣ x ∣ . f ( x ) = x 4 − x 2 , f ( x ) = ∣ x ∣
Explanation
Understanding Even Functions We are given four functions and we need to determine which of them are even. A function f ( x ) is even if f ( − x ) = f ( x ) for all x in its domain. We will check each function separately.
Analyzing f(x) = x^4 - x^2 Let's analyze the first function: f ( x ) = x 4 − x 2 . We need to find f ( − x ) and see if it's equal to f ( x ) .
f ( − x ) = ( − x ) 4 − ( − x ) 2 = x 4 − x 2
Since f ( − x ) = x 4 − x 2 = f ( x ) , the function f ( x ) = x 4 − x 2 is even.
Analyzing f(x) = x^2 - 3x + 2 Now let's analyze the second function: f ( x ) = x 2 − 3 x + 2 . We need to find f ( − x ) and see if it's equal to f ( x ) .
f ( − x ) = ( − x ) 2 − 3 ( − x ) + 2 = x 2 + 3 x + 2
Since f ( − x ) = x 2 + 3 x + 2 e q x 2 − 3 x + 2 = f ( x ) , the function f ( x ) = x 2 − 3 x + 2 is not even.
Analyzing f(x) = sqrt(x-2) Now let's analyze the third function: f ( x ) = x − 2 . We need to find f ( − x ) and see if it's equal to f ( x ) .
f ( − x ) = − x − 2
For f ( x ) to be even, we need − x − 2 = x − 2 for all x in the domain. However, the domain of f ( x ) = x − 2 is x ≥ 2 , and the domain of f ( − x ) = − x − 2 is x ≤ − 2 . Since the domains are different, and f ( − x ) = f ( x ) , the function f ( x ) = x − 2 is not even.
Analyzing f(x) = |x| Finally, let's analyze the fourth function: f ( x ) = ∣ x ∣ . We need to find f ( − x ) and see if it's equal to f ( x ) .
f ( − x ) = ∣ − x ∣ = ∣ x ∣
Since f ( − x ) = ∣ x ∣ = f ( x ) , the function f ( x ) = ∣ x ∣ is even.
Conclusion Therefore, the even functions are f ( x ) = x 4 − x 2 and f ( x ) = ∣ x ∣ .
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 understand phenomena. For example, in signal processing, even functions represent signals that are time-symmetric, which can simplify the analysis and design of filters. In architecture, symmetrical designs often incorporate even functions to ensure visual balance and structural stability. Understanding even functions helps in modeling and predicting behavior in various symmetric systems.
