SOLUTION: How do you determine whether {{{ f(x) = (X^2+1)/(x)}}} is an even, odd, or neither and what is its symmetry? If I recall correctly even tests are f(-x) = f(x), odd tests are f(-

Algebra ->  Functions -> SOLUTION: How do you determine whether {{{ f(x) = (X^2+1)/(x)}}} is an even, odd, or neither and what is its symmetry? If I recall correctly even tests are f(-x) = f(x), odd tests are f(-      Log On


   

Question 1121642: How do you determine whether +f%28x%29+=+%28X%5E2%2B1%29%2F%28x%29 is an even, odd, or neither and what is its symmetry?
If I recall correctly even tests are f(-x) = f(x), odd tests are f(-x) = -f(x) and even has a symmetry in respects to the y-axis and odd to the origin? But I don't know how to approach the above function.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

even tests are+f%28-x%29+=+f%28x%29,
given: f%28x%29=%28x%5E2%2B1%29%2Fx+
f%28-x%29=%28%28-x%29%5E2%2B1%29%2F%28-x%29
f%28-x%29+=+-%28x%5E2%2B1%29%2Fx+=> +f%28-x%29+=+-f%28x%29
so, f%28-x%29+%3C%3E+f%28x%29++=> f%28x%29=%28x%5E2%2B1%29%2Fx is not an even function

odd tests are f%28-x%29+=+-f%28x%29
since above is proven that f(-x) = -f(x), means f(x) = (x^2 + 1)/x is an odd function

Since the function is +odd, it is symmetric about the origin.

+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28x%5E2%2B1%29%2Fx%29+


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


%28x%5E2%2B1%29%2Fx = x+%2B+1%2Fx

Both x and 1/x are odd functions; the sum of two or more odd functions is an odd function.

Therefore, %28x%5E2%2B1%29%2Fx%29 is an odd function.