Answer:
discuss this with your teacher to find their reasoning
f(x) is an even function
Explanation:
You want to know if f(x) = (x⁵ -x)/x³ is even or odd.
Even function
A function is even if ...
f(-x) = f(x)
Here, we have ...
f(x) = (x⁵ -x)/x³
f(-x) = ((-x)⁵ -(-x))/(-x)³ = (-x⁵ +x)/-x³ = (x⁵ -x)/x³
This is identical to f(x), so we can conclude this function is even. The graph of an even function is symmetrical about the y-axis, as is the graph in the attachment.
Odd function
A function is odd if ...
f(-x) = -f(x)
As we saw above, f(-x) = (x⁵ -x)/x³, which is not -f(x). This function is not odd.
The graph of an odd function is symmetrical about the origin.
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Additional comment
Your teacher may be reacting to the fact that all of the exponents are odd numbers (1, 3, 5). If this were a polynomial, not a rational function, the odd exponents would signify it is an odd function.
Because it is the ratio of odd functions. the oddness of the numerator is cancelled by the oddness of the denominator, leaving an even function. It is not so different from the product f(x) = (x^-3)(x^5 -x). This, as does the original form of f(x), reduces to x^2 -x^-2, the sum of even functions of x.