Are the images of odd functions crossing the origin? Do the images of even functions intersect the y-axis

Are the images of odd functions crossing the origin? Do the images of even functions intersect the y-axis

For example, y = 1 / X is an odd function, but it is not the origin
Even functions, too, may not intersect the y-axis, for example, y = x squared + 1

Given the vector a = (SIN3 / x, cos3 / x), B = (cos3 / x, root 3cos3 / x), the function f (x) = vector a · vector B. (1) find the monotone increasing interval of function f (x) (2) if △ ABC's three sides, a, B, C satisfy B & sup2; = AC, and the angle of B is x, try to find the range of X and the range of function f (x)

1. Given vector a = (SIN3 / x, cos3 / x), B = (cos3 / x, root 3cos3 / x), function f (x) = vector a · vector B
Then f (x) = sin (3 / x) cos (3 / x) + cos (3 / x) * √ 3 * cos (3 / x)
=(1 / 2) * sin (2x / 3) + (√ 3 / 2) * [cos (2x / 3) + 1]
=Sin [(2x / 3) + π / 3] + √ 3 / 2
Then when 2K π - π ≤ (2x / 3) + π / 3 ≤ 2K π, that is, 3K π - 2 π ≤ x ≤ 3K π - π / 2, K ∈ Z, the function f (x) is an increasing function
So the monotone increasing interval of function f (x) is [3K π - 2 π, 3K π - π / 2], K ∈ Z
2. Given that the angle of edge B is x, then:
The cosine theorem is: cosx = (A & # 178; + C & # 178; - B & # 178;) / (2Ac)
And B & # 178; = AC, so:
cosx=(a²+c²-ac)/(2ac)
From the mean value theorem a & # 178; + C & # 178; ≥ 2Ac (take the equal sign if and only if a = C)
Then (A & # 178; + C & # 178; - AC) / (2Ac) ≥ AC / (2Ac) = 1 / 2
That is, cosx ≥ 1 / 2
The solution is 0

Given a = (cos3 / 2x, SIN3 / 2x), B = (cosx / 2, - SiNx / 2), if f (x) = a * B (1), find the minimum period of function f (x)
(2) If x ∈ [- π / 3, π / 4], find the maximum and minimum of the function

a*b=cos3/2x*cosx/2+sin3/2x*（-sinx/2）=cos(3/2x+x/2)=cos2x,
Minimum period T = 2 π / 2 = π