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.
$g(x)=x^4-3 x$
Determine whether the function is even, odd, or neither. Choose the correct answer below.
Answer (2)
Calculate g ( − x ) : g ( − x ) = ( − x ) 4 − 3 ( − x ) = x 4 + 3 x .
Check if g ( − x ) = g ( x ) (even): x 4 + 3 x e q x 4 − 3 x , so g ( x ) is not even.
Check if g ( − x ) = − g ( x ) (odd): x 4 + 3 x e q − x 4 + 3 x , so g ( x ) is not odd.
Conclude that g ( x ) is neither even nor odd, and its graph has neither y -axis nor origin symmetry: neither .
Explanation
Problem Analysis We are given the function g ( x ) = x 4 − 3 x . Our goal is to determine whether this function is even, odd, or neither, and to identify the symmetry of its graph.
Definitions of 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. The graph of an even function is symmetric with respect to the y -axis, and the graph of an odd function is symmetric with respect to the origin.
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 , 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 , so the function is not odd.
Determining Symmetry Since the function is neither even nor odd, its graph has neither y -axis nor origin symmetry.
Conclusion Therefore, the function g ( x ) = x 4 − 3 x is neither even nor odd, and its graph has neither y -axis nor origin symmetry.
Examples
Understanding whether a function is even or odd helps in simplifying complex calculations in physics and engineering. For example, when analyzing symmetrical systems, recognizing even or odd functions can reduce the computational effort required to model the system's behavior. In signal processing, even and odd functions are used to decompose signals into simpler components, aiding in signal analysis and filtering. This classification also helps in predicting the behavior of systems under certain transformations, making it a valuable tool in various scientific and engineering applications.
The function g ( x ) = x 4 − 3 x is neither even nor odd, as it does not satisfy the conditions for evenness or oddness. Therefore, its graph has neither y -axis nor origin symmetry. The answer is neither .
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