For these functions determine which ones are EVEN (y axis symmetry). a) f(x) = 3x^4 + 4 b) f(x) = x^3 + 4 c) f(x) = x^2 + x - 3 d) f(x) = 2x^4 + x^2 + 1 e) f(x) = 4 f) f(x) = x^4 + x^3 + x^2 + x+ 1
A function is even if `f(-x)=f(x)` :
(a) `f(x)=3x^4+4` `f(-x)=3(-x)^4+4=3x^4+4` even
(b) `f(x)=x^3+4` `f(-x)=(-x)^3+4=-x^3+4`
(c) `f(x)=x^2+x-3` `f(-x)=(-x)^2+(-x)-3=x^2-x-3` not even
(d) `f(x)=2x^4+x^2+1` `f(-x)=2(-x)^4+(-x)^2+1=2x^4+x^2+1` even
(e) `f(x)=4` `f(-x)=4` even
(f) `f(x)=x^4+x^3+x^2+x+1` `f(-x)=(-x)^4+(-x)^3+(-x)^2+(-x)+1=x^4-x^3+x^2-x+1` not even
Note that for polynomials, it suffices that all exponents be even. Also note that not even does not mean odd.
For every function marked as even, the function is symmetric about the y-axis. Here are their graphs:
If we have a function where f(x) = f(-x), the function is said to be even.
In mathematical terms, even functions are said to be reflexive (or symmetric) about the y-axis.
Therefore, in order to test whether a function is even or not, put –x for x in f(x)and test whether f(x) is returned or not.
When all the exponents of x in f(x) are even (including 0), the function is even. If any of the exponents is odd, the function is not even.
Thus in example a),
f(x) = 3x^4 + 4
f(-x) = 3(-x)^4 + 4 = 3x^4 + 4 = f(x);
Hence it is even.
Testing in similar arguments it is clear that functions a), d) and e) are even.