Respuesta :
Answer:
even
Step-by-step explanation:
f(-x)=f(x) implies the function is even.
f(-x)=-f(x) implies the function is odd.
There are some things you should know before we being this problem:
[tex](\text{negative number})^\text{even}=\text{positive number}[/tex]
[tex](\text{negative number})^\text{ odd }=\text{ negative number}[/tex]
Examples:
[tex](-x)^{22}=x^{22}[/tex]
[tex](-x)^{23}=-x^{23}[/tex]
Let's begin the problem:
[tex]f(x)=3x^4[/tex]
Must plug in -x. Both definitions require this part.
[tex]f(-x)=3(-x)^4[/tex]
[tex]f(-x)=3(x^4)[/tex]
[tex]f(-x)=3x^4[/tex]
[tex]f(-x)=f(x)[/tex] I replace [tex]3x^4 \text{ with }f(x)\text{ because } f(x)=3x^4[/tex]
So since we have f(-x)=f(x), the conclusion is that f is even.
to test a function's behaviour if it's even or odd, we simply use -x, namely check what f(-x) give us, if the result is the same as the original f(x), then it's even, if the result is a negative original, namely -f(x), then is odd.
[tex]\bf f(-x)=3(-x)^4\implies f(-x)=3(-x)(-x)(-x)(-x)\implies f(-x)=\underset{\underset{even}{\uparrow }}{\stackrel{\stackrel{f(x)}{\downarrow }}{3x^4}}[/tex]
