How do you determine if #f(x)= x^4 - 4x^2# is an even or odd function?

Question

Chloe Alexander · Accepted Answer

#f(x)" is even"# a function is odd if #f(-x)=-f(x)# a function is even if #f(-x)=f(x)# #f(x)=x^4-4x^2# #f(-x)=(-x)^4-4(-x)^2# #f(-x)=x^4-4x^2=f(x)# #:. f(x)" is even"#

Lily Anderson · Answer

undefined To determine if a function $f(x) = x^4 - 4x^2$ is even or odd, we can use the properties of even and odd functions.

1. Even Function: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain of $f$.

2. Odd Function: A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$.

Let's check both conditions for $f(x) = x^4 - 4x^2$:

1. Even Function:
   Substitute $-x$ into $f(x)$: $f(-x) = (-x)^4 - 4(-x)^2$
   Simplify: $f(-x) = x^4 - 4x^2$
   
   Since $f(-x) = f(x)$ for all $x$, the function $f(x) = x^4 - 4x^2$ is an even function.

2. Odd Function:
   Substitute $-x$ into $f(x)$: $f(-x) = (-x)^4 - 4(-x)^2$
   Simplify: $f(-x) = x^4 - 4x^2$

Since $f(-x) = f(x)$ and not $-f(x)$ for all $x$, the function $f(x) = x^4 - 4x^2$ is not an odd function.

Therefore, $f(x) = x^4 - 4x^2$ is an even function.