Problem

Determine whether the function f(x)=x4−4x2+1 is symmetric about the y-axis, the origin, or neither.

Answer

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Answer

Step 2: Check symmetry about the origin by replacing x with -x in the function. If the function changes sign, then the function is symmetric about the origin. So, f(−x)=(−x)4−4(−x)2+1=x4−4x2+1=f(x). Therefore, the function is not symmetric about the origin because it does not change sign when x is replaced with -x.

Steps

Step 1 ：Step 1: Check symmetry about the y-axis by replacing x with -x in the function. If the function remains the same, then the function is symmetric about the y-axis. So, f(−x)=(−x)4−4(−x)2+1=x4−4x2+1=f(x). Therefore, the function is symmetric about the y-axis.

Step 2 ：Step 2: Check symmetry about the origin by replacing x with -x in the function. If the function changes sign, then the function is symmetric about the origin. So, f(−x)=(−x)4−4(−x)2+1=x4−4x2+1=f(x). Therefore, the function is not symmetric about the origin because it does not change sign when x is replaced with -x.

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