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17 Feb 2010, 00:08
Kudos
Bookmarks
You may be asked to "determine algebraically" whether a function is even or odd. To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(–x) = f(x), so all of the signs are the same), then the function is even. If you end up with the exact opposite of what you started with (that is, if f(–x) = –f(x), so all of the "plus" signs become "minus" signs, and vice versa), then the function is odd. So I'll plug –x in for x, and simplify: In all other cases, the function is "neither even nor odd".
Eg 1: Determine algebraically whether f(x) = –3x^2 + 4 is even, odd, or neither.
So I'll plug –x in for x, and simplify:
f(–x) = –3(–x)^2 + 4
= –3(x^2) + 4
= –3x^2 + 4
My final expression is the same thing I'd started with, which means that f(x) is even
Eg 2: Determine algebraically whether f(x) = 2x^3 – 4x is even, odd, or neither.
so I'll plug –x in for x, and simplify:
f(–x) = 2(–x)^3 – 4(–x)
= 2(–x^3) + 4x
= –2x^3 + 4x
My final expression is the exact opposite of what I started with, by which I mean that the sign on each term has been changed to its opposite, just as if I'd multiplied through by –1:
–f(x) = –1[f(x)]
= –[2x^3 – 4x]
= –2x^3 + 4x
This means that f(x) is odd.
Eg3 : Determine algebraically whether f(x) = 2x^3 – 3x^2 – 4x + 4 is even, odd, or neither.
i'll plug –x in for x, and simplify:
f(–x) = 2(–x)^3 – 3(–x)^2 – 4(–x) + 4
= 2(–x^3) – 3(x^2) + 4x + 4
= –2x^3 –3x^2 + 4x + 4
This is neither the same thing I started with (namely, 2x^3 – 3x^2 – 4x + 4) nor the exact opposite of what I started with (namely, –2x^3 + 3x^2 + 4x – 4). This means that
f(x) is neither even nor odd.
Eg 1: Determine algebraically whether f(x) = –3x^2 + 4 is even, odd, or neither.
So I'll plug –x in for x, and simplify:
f(–x) = –3(–x)^2 + 4
= –3(x^2) + 4
= –3x^2 + 4
My final expression is the same thing I'd started with, which means that f(x) is even
Eg 2: Determine algebraically whether f(x) = 2x^3 – 4x is even, odd, or neither.
so I'll plug –x in for x, and simplify:
f(–x) = 2(–x)^3 – 4(–x)
= 2(–x^3) + 4x
= –2x^3 + 4x
My final expression is the exact opposite of what I started with, by which I mean that the sign on each term has been changed to its opposite, just as if I'd multiplied through by –1:
–f(x) = –1[f(x)]
= –[2x^3 – 4x]
= –2x^3 + 4x
This means that f(x) is odd.
Eg3 : Determine algebraically whether f(x) = 2x^3 – 3x^2 – 4x + 4 is even, odd, or neither.
i'll plug –x in for x, and simplify:
f(–x) = 2(–x)^3 – 3(–x)^2 – 4(–x) + 4
= 2(–x^3) – 3(x^2) + 4x + 4
= –2x^3 –3x^2 + 4x + 4
This is neither the same thing I started with (namely, 2x^3 – 3x^2 – 4x + 4) nor the exact opposite of what I started with (namely, –2x^3 + 3x^2 + 4x – 4). This means that
f(x) is neither even nor odd.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
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17 Feb 2010, 14:37
Kudos
Bookmarks
Thanks. But that green color gives tough time while reading.. 
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
