Question

Determine algebraically whether the function is even, odd, or neither even nor odd. f as a function of x is equal to x plus quantity 4 over x.

Answer 1

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 $f(x)=x+\frac{4}{x}$ is even, odd, or neither.

2) Determine if $f(x)=\frac{x+4}{x}$ is even, odd, or neither.

Step-by-step explanation:

$f(-x)=f(x)$ implies $f$ is even.

$f(-x)=-f(x)$ implies $f$ is odd.

So either definition says we have to plug in $-x$.

1)

$f(x)=x+\frac{4}{x}$ with new input $-x$:

$f(-x)=-x+\frac{4}{-x}$

$f(-x)=-x+-\frac{4}{x}$

$f(-x)=-(x+\frac{4}{x})$

$f(-x)=-f(x)$

This means $f$ is even since we got the same thing we started with.

2)

$f(x)=\frac{x+4}{x}$ with new input $-x$:

$f(-x)=\frac{-x+4}{-x}$

$f(-x)=\frac{-(x-4)}{-x}$

$f(-x)=\frac{x-4}{x}$

This is neither the same or the opposite of what we started with.