Question

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

part 2 of 3
(b) is ( f(-x) = f(x) )?
yes, ( f(-x) = f(x) )

part: 2 / 3
part 3 of 3
(c) is this function even, odd, or neither?
since ( f(-x) = f(x) ), the function is select

Explanation:

Part 1 of 3

Step1: Substitute -x into f(x)

To find \( f(-x) \), we substitute \( -x \) for \( x \) in the function \( f(x) = -x^4 + 4|x| \). So we get \( f(-x) = -(-x)^4 + 4|(-x)| \).

Step2: Simplify the expressions

First, simplify \( (-x)^4 \). Since any even power of a negative number is positive, \( (-x)^4 = x^4 \), so \( -(-x)^4 = -x^4 \). Then, simplify \( |-x| \). The absolute value of a negative number is its positive counterpart, so \( |-x| = |x| \). Therefore, \( 4|(-x)| = 4|x| \). Combining these, we have \( f(-x) = -x^4 + 4|x| \).

We compare \( f(-x) \) (which we found to be \( -x^4 + 4|x| \)) with \( f(x) \) (given as \( -x^4 + 4|x| \)). Since both expressions are identical, \( f(-x) = f(x) \).

A function is even if \( f(-x) = f(x) \) for all \( x \) in the domain, and odd if \( f(-x) = -f(x) \) for all \( x \) in the domain. We already determined that \( f(-x) = f(x) \), so by the definition of an even function, this function is even.

Answer:

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

Part 2 of 3