Which of the following is an even function?
[tex]f(x)=|x|[/tex]
[tex]f(x)=x^3-1[/tex]
[tex]f(x)=-3 x[/tex]
[tex]f(x)=\sqrt[3]{x}[/tex]
Answer (2)
An even function satisfies the condition f ( − x ) = f ( x ) .
Test f ( x ) = ∣ x ∣ : f ( − x ) = ∣ − x ∣ = ∣ x ∣ = f ( x ) . Thus, f ( x ) = ∣ x ∣ is even.
Test the other functions and find that they are not even.
The only even function is f ( x ) = ∣ x ∣ .
Explanation
Understanding Even Functions We are given four functions and need to determine which one is even. 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| Let's test f ( x ) = ∣ x ∣ . We have f ( − x ) = ∣ − x ∣ . Since the absolute value of a number is the same as the absolute value of its negative, we have ∣ − x ∣ = ∣ x ∣ . Therefore, f ( − x ) = ∣ x ∣ = f ( x ) , so f ( x ) = ∣ x ∣ is an even function.
Testing f(x) = x^3 - 1 Now let's test f ( x ) = x 3 − 1 . We have f ( − x ) = ( − x ) 3 − 1 = − x 3 − 1 . Since − x 3 − 1 e q x 3 − 1 , f ( x ) = x 3 − 1 is not an even function.
Testing f(x) = -3x Next, let's test f ( x ) = − 3 x . We have f ( − x ) = − 3 ( − x ) = 3 x . Since 3 x e q − 3 x , f ( x ) = − 3 x is not an even function.
Testing f(x) = cube root of x Finally, let's test f ( x ) = 3 x . We have f ( − x ) = 3 − x = − 3 x . Since − 3 x e q 3 x , f ( x ) = 3 x is not an even function.
Conclusion Therefore, the only even function among the given options is f ( x ) = ∣ x ∣ .
Examples
Even functions are symmetric with respect to the y-axis. This property is useful in physics and engineering, where symmetry can simplify calculations. For example, when analyzing the vibrations of a symmetrical structure, even functions can describe the symmetric modes of vibration, making the analysis easier. Similarly, in signal processing, even functions are used to represent signals that have symmetry, which simplifies their analysis and manipulation.
The only even function among the given options is f ( x ) = ∣ x ∣ , as it satisfies the condition f ( − x ) = f ( x ) . The other functions do not meet this criterion. Therefore, the answer is f ( x ) = ∣ x ∣ .
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