An even function is a function such that f(-x)=f(x)
An odd fucntion is a function such that f(-x)=-f(x)
Additionally..Even functions have all even powers..Odd functions have all odd powers
With that said...The book indicates that F(x)=1 is an EVEN fucntion???
1 is 1^1 and 1 is also 1^2..OR 1^whateryouwant
I am just not seeing this one. Help!
2007-04-20 19:46:16 · 6 answers · asked by RScott 3 in Science & Mathematics ➔ Mathematics
Whoa..Hold the phone..Is this even because this is the graph of a horizontal line..WHich means it symmetrical to the Y axis..Which means it is by definition Even?
2007-04-20 19:47:47 · update #1
Here, we are not raising 1 to any power, we are raising x to a power. In this case, f(x)=1 can be translated to f(x)=x^0, x raised to the zero power. Recall that any number raised to the zero power equals one. Plug in any number for x in the equation and you get 1. 0 is defined as an even number, so f(x)=x^0 disguised as f(x)=1 is raised to an even power, and thus is an even function.
2007-04-20 19:57:26 · answer #1 · answered by Supermatt100 4 · 1⤊ 0⤋
Yes, f(x)=1 is an even function. The descriptions of even and odd functions are based on the properties of polynomials -- if P is a polynomial where only even powers of the variable appear, then P(x) = P(-x), which is the definition of an even function. Similarly, if P is a polynomial where only odd powers of the variable appear, then P(x) = -P(-x), which is the definition of an odd function. That's what your book is referring to when it states "Even functions have all even powers..Odd functions have all odd powers" (even though an even or odd function may not be a polynomial at all and thus lack any "powers" of x to be had.
That said, f(x)=1 obeys the more restrictive characterization as well, because 1=x^0, and 0 is an EVEN power of x. It is not the case that 1=x^1 or 1=x^2 or any power you want, unless of course you only allow x to be 1.
2007-04-20 19:57:50 · answer #2 · answered by Pascal 7 · 1⤊ 0⤋
The effect of even-ness is in fact that the y-axis serves as an axis of symmetry for the function.
The effect of odd-ness is point symmetry about the origin - if (4,3) is on the function, so is (-4,-3), if (-2, 7) is on the function, so is (2, -7)
2007-04-20 20:04:27 · answer #3 · answered by lockedjew 5 · 1⤊ 0⤋
You got it. Also f(x) = 1, f(-x) = 1 so f(x) = f(-x), meeting the definition of an even function
2007-04-20 19:52:07 · answer #4 · answered by gp4rts 7 · 1⤊ 0⤋
f(x)=1 is even because for all values of -x there is an equal found at x. f(x) = 1 isnt very interesting though.
2007-04-20 19:51:12 · answer #5 · answered by James H 2 · 1⤊ 0⤋
yes .. it doesn't depend on x so f(x)=f(-x) and its even
2007-04-20 19:50:50 · answer #6 · answered by hustolemyname 6 · 1⤊ 0⤋