1.6. Symmetry http://www.ck12.org
Answers:
- Even Functions: The squaring function, the absolute value function.
Odd Functions: The identity function, the cubing function, the reciprocal function, the sine function.
Neither: The square root function, the exponential function and the log function. The logistic function is also neither
because it is rotationally symmetric about the point( 0 ,^12 )as opposed to the origin. - Ifh(x)is even thenh(−x) =h(x). Ifg(x)is odd theng(−x) =−g(x).
Therefore:f(−x) =h(−x)+g(−x) =h(x)−g(x)
This does not matchf(x) =h(x)+g(x)nor does it match−f(x) =−h(x)−g(x).
This is a proof that shows the sum of an even function and an odd function will never itself be even or odd.
f(x) =x(x^2 − 1 )(x^4 + 1 )
f(−x) = (−x)((−x)^2 − 1 )((−x)^4 + 1 )
=−x(x^2 − 1 )(x^4 + 1 )
=−f(x)The function is odd.
Practice
Determine whether the following functions are even, odd, or neither.
1.f(x) =− 4 x^2 + 1
2.g(x) = 5 x^3 − 3 x
3.h(x) = 2 x^2 −x
4.j(x) = (x− 4 )(x− 3 )^3
5.k(x) =x(x^2 − 1 )^2
6.f(x) = 2 x^3 − 5 x^2 − 2 x+ 1
7.g(x) = 2 x^2 − 4 x+ 2
8.h(x) =− 5 x^4 +x^2 + 2
- Supposeh(x)is even andg(x)is odd. Show thatf(x) =h(x)−g(x)is neither even nor odd.
- Supposeh(x)is even andg(x)is odd. Show thatf(x) =hg((xx))is odd.
- Supposeh(x)is even andg(x)is odd. Show thatf(x) =h(x)·g(x)is odd.