Question

part 1 of 3
(a) given ( f(x) = -x^4 + 4|x| ), find ( f(-x) ).
( f(-x) = -(-x)^4 + 4|(-x)| )
( = square )

Explanation:

Step1: Simplify \((-x)^4\)

Recall that for any real number \(a\) and positive integer \(n\), \((-a)^n=a^n\) when \(n\) is even. Here \(n = 4\) (even), so \((-x)^4=x^4\). Then \(-(-x)^4=-x^4\).

Step2: Simplify \(| - x|\)

Recall the property of absolute value: \(| - a|=|a|\) for any real number \(a\). So \(| - x| = |x|\), and \(4| - x|=4|x|\).

Step3: Combine the results

Substitute the simplified forms back into the expression for \(f(-x)\): \(f(-x)=-x^4 + 4|x|\).

Answer:

\(-x^4 + 4|x|\)