i'm a secondary school student and i've faced this problem :
f is an odd function defined on R. f(x) = g(x) if x ∈ ]-∞; 0 [ and g(x) is a function defined on R.
g(x) = (x^2 + 4x +3) / (2x^2 + 8x +9).
-1 is the minium of g and g(x) < 1/2 but 1/2 isn't the maximum of g.
This is also additional information that should help:
f(-3) = 0
f(-1) = 0
f(1) = 0
f(-2) = -1
f(3) = 0
Now, i'm asked to find f(0) (analytically) and to complete the curve of f.
In this image there is the curve of it that I generated from a website. However, in the exercice, i'm just given the part of the curve of f that is in the interval ]- ∞;0[ and i'm asked to JUSTIFY the solution : image
##NOTE : ##
f(x) isn't the same in the positive interval as shown in the image
Since f is odd you can easily deduce the positive part of the curve.
The positive part of f is the symmetric through O of the negative part
So for x>0, f(x)=-f(-x)=-g(-x).
And f(0)=0 since f(0)=-f(-0)=-f(0).