Determine whether the function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the $y$-axis, the origin, or neither.
$g(x)=x^4-3 x$
Answer (2)
Calculate g ( − x ) : g ( − x ) = ( − x ) 4 − 3 ( − x ) = x 4 + 3 x .
Check if g ( − x ) = g ( x ) : x 4 + 3 x e q x 4 − 3 x , so g ( x ) is not even.
Check if g ( − x ) = − g ( x ) : x 4 + 3 x e q − x 4 + 3 x , so g ( x ) is not odd.
The function is neither even nor odd, and the graph has neither y -axis nor origin symmetry. neither
Explanation
Problem Analysis We are given the function g ( x ) = x 4 − 3 x . We need to determine if this function is even, odd, or neither. Then, we need to determine the symmetry of the graph of the function.
Even and Odd Functions A function is even if g ( − x ) = g ( x ) for all x . A function is odd if g ( − x ) = − g ( x ) for all x . If neither of these conditions is met, the function is neither even nor odd.
Calculating g(-x) Let's find g ( − x ) by substituting − x for x in the expression for g ( x ) : g ( − x ) = ( − x ) 4 − 3 ( − x ) = x 4 + 3 x .
Checking for Even Function Now, let's compare g ( − x ) with g ( x ) . We have g ( x ) = x 4 − 3 x and g ( − x ) = x 4 + 3 x . Since x 4 + 3 x e q x 4 − 3 x , we conclude that g ( − x ) e q g ( x ) . Therefore, the function is not even.
Checking for Odd Function Next, let's check if g ( − x ) = − g ( x ) . We have − g ( x ) = − ( x 4 − 3 x ) = − x 4 + 3 x . Comparing this with g ( − x ) = x 4 + 3 x , we see that x 4 + 3 x e q − x 4 + 3 x . Therefore, g ( − x ) e q − g ( x ) , and the function is not odd.
Symmetry Since the function is neither even nor odd, the graph of the function is not symmetric with respect to the y -axis or the origin.
Examples
Understanding whether a function is even or odd helps in simplifying complex mathematical models in physics and engineering. For example, in signal processing, even functions represent signals that are symmetric in time, while odd functions represent signals that are anti-symmetric. Recognizing these symmetries can significantly reduce the computational effort required to analyze these signals. Similarly, in structural mechanics, understanding symmetry can simplify the analysis of stress distribution in symmetric structures, making calculations more manageable and efficient. The function's symmetry properties can be used to predict its behavior and simplify calculations.
The function g ( x ) = x 4 − 3 x is neither even nor odd. Its graph is not symmetric with respect to the y -axis or the origin. The conclusion is that the function is classified as neither.
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