Which statement best describes how to determine whether $f(x)=x^4-x^3$ is an even function?
A. Determine whether $(-x)^4-(-x)^3$ is equivalent to $x^4-x^3$.
B. Determine whether $\left(-x^4\right)-\left(-x^3\right)$ is equivalent to $x^4+x^3$.
C. Determine whether $(-x)^4-(-x)^3$ is equivalent to $-\left(x^4-x^3\right)$.
D. Determine whether $\left(-x^4\right)-\left(-x^3\right)$ is equivalent to $-\left(x^4+x^3\right)$.
Answer (2)
To check if f ( x ) is even, verify if f ( − x ) = f ( x ) .
Evaluate f ( − x ) = ( − x ) 4 − ( − x ) 3 = x 4 + x 3 .
Compare f ( − x ) = x 4 + x 3 with f ( x ) = x 4 − x 3 .
Determine whether ( − x ) 4 − ( − x ) 3 is equivalent to x 4 − x 3 . Determine whether ( − x ) 4 − ( − x ) 3 is equivalent to x 4 − x 3
Explanation
Understanding Even Functions To determine if a function f ( x ) is even, we need to check if f ( − x ) = f ( x ) for all x in the domain of f . In this case, f ( x ) = x 4 − x 3 . We need to evaluate f ( − x ) and see if it is equal to x 4 − x 3 .
Evaluating f(-x) Let's find f ( − x ) by substituting − x for x in the expression for f ( x ) : f ( − x ) = ( − x ) 4 − ( − x ) 3
Simplifying f(-x) Now, let's simplify the expression. Recall that ( − x ) 4 = x 4 because a negative number raised to an even power is positive. Also, ( − x ) 3 = − x 3 because a negative number raised to an odd power is negative. Therefore, f ( − x ) = x 4 − ( − x 3 ) = x 4 + x 3
Comparing f(-x) and f(x) Now we compare f ( − x ) = x 4 + x 3 with f ( x ) = x 4 − x 3 . Since x 4 + x 3 is not equal to x 4 − x 3 , the function f ( x ) = x 4 − x 3 is not an even function.
Analyzing the Statements Now let's examine the given statements to see which one correctly describes the process of determining if f ( x ) is even. The correct statement should involve checking if ( − x ) 4 − ( − x ) 3 is equivalent to x 4 − x 3 .
Conclusion The first statement is: Determine whether ( − x ) 4 − ( − x ) 3 is equivalent to x 4 − x 3 .
Since f ( − x ) = ( − x ) 4 − ( − x ) 3 = x 4 + x 3 and f ( x ) = x 4 − x 3 , we are checking if x 4 + x 3 = x 4 − x 3 . This is the correct way to determine if the function is even.
Examples
Even functions are symmetric about the y-axis. In physics, the potential energy function of a simple harmonic oscillator, V ( x ) = 2 1 k x 2 , is an even function, meaning the potential energy is the same for equal displacements on either side of the equilibrium point. Similarly, in signal processing, even signals have symmetric waveforms, which simplifies their analysis and processing. Recognizing even functions helps simplify calculations and understand symmetries in various scientific and engineering contexts.
To determine if f ( x ) = x 4 − x 3 is even, we check if f ( − x ) = f ( x ) . After evaluating, we find that f ( − x ) = x 4 + x 3 is not equal to f ( x ) = x 4 − x 3 , showing the function is not even. Therefore, the best statement is option A.
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