even or odd function

[sanchez]

Is there a way to tell whether a function such as this one is odd or even just by looking at it? Thank you.

f(x) = -5x^5 + x^3

 

[royhaas]

Yes, if you have enough experience.

 

[Denis]

Is there a way to tell whether a function such as this one is odd or even just by looking at it? Thank you.
f(x) = -5x^5 + x^3

-5x^5 ends with 0 is x even, with 5 if x odd: that's all you need to notice.

 

[Matt]

Hello Mr. Sanchez,

Is there a way to tell whether a function such as this one is odd or even just by looking at it? Thank you.

For polynomials, if all the exponents are even, then the function is even. If all the exponents are odd, then the function is odd.

For example, f(x) = x^2 + 5x^8 is even,
whereas g(x) = 8231x^13 + 3x^7 is odd.

f(x) = -5x^5 + x^3

This polynomial contains both even and odd exponents and is therefore neither even nor odd.

 

[stapel]

This polynomial [f(x) = -5x⁵ + x³] contains both even and odd exponents....

I'm pretty sure both "5" and "3" are odd. :wink:

Eliz.

 

[Mrspi]

This polynomial [f(x) = -5x⁵ + x³] contains both even and odd exponents....

I'm pretty sure both "5" and "3" are odd. :wink:

Eliz.

I agree, Eliz......

Since the coefficient of the x^0 term is 0, this polynomial contains only odd exponents and thus is an odd function.

We could go back to the definition of an odd function. f(x) is an odd function if and only if
f(-x) = - f(x)

Replacing x with (-x) in the function gives us
f(-x) = -5(-x)^5 + (-x)^3
f(-x) = -5(-x^5) + (-x^3)
f(-x) = 5x^5 - x^3
f(-x) = -(-5x^5 + x^3)
f(-x) = -f(x)

 

[Matt]

This polynomial [f(x) = -5x⁵ + x³] contains both even and odd exponents....

I'm pretty sure both "5" and "3" are odd. :wink:

Indeed! How in blazes did I miss that? :shock: