Respuesta :
Answer:
1) Even for first problem.
2) Neither for the second.
I really think the problem I labeled 1 is the correct interpretation but just in case you meant the latter I wrote the latter as well. The sentence translates to exactly what I have for problem number 1.
The Problem:
1) Determine if [tex]f(x)=x+\frac{4}{x}[/tex] is even, odd, or neither.
2) Determine if [tex]f(x)=\frac{x+4}{x}[/tex] is even, odd, or neither.
Step-by-step explanation:
[tex]f(-x)=f(x)[/tex] implies [tex]f[/tex] is even.
[tex]f(-x)=-f(x)[/tex] implies [tex]f[/tex] is odd.
So either definition says we have to plug in [tex]-x[/tex].
1)
[tex]f(x)=x+\frac{4}{x}[/tex] with new input [tex]-x[/tex]:
[tex]f(-x)=-x+\frac{4}{-x}[/tex]
[tex]f(-x)=-x+-\frac{4}{x}[/tex]
[tex]f(-x)=-(x+\frac{4}{x})[/tex]
[tex]f(-x)=-f(x)[/tex]
This means [tex]f[/tex] is even since we got the same thing we started with.
2)
[tex]f(x)=\frac{x+4}{x}[/tex] with new input [tex]-x[/tex]:
[tex]f(-x)=\frac{-x+4}{-x}[/tex]
[tex]f(-x)=\frac{-(x-4)}{-x}[/tex]
[tex]f(-x)=\frac{x-4}{x}[/tex]
This is neither the same or the opposite of what we started with.
