f(x)= x/( 1+x)=( 1+x- 1)/( 1+x)= 1- 1/( 1+x)
Pull f(x)=- 1/x to shift 1 unit to the left, and then shift 1 unit to get f(x)=x/( 1+x).
draw
()
(2) Solution: Make the function meaningful
Just 1+x≠0
∴x≠- 1
Let x 1 < x2 (x 1, x2≠0), then
f(x2)-f(x 1)
= x2/( 1+x2)-x 1/( 1+x 1)
=[x2( 1+x 1)-x 1( 1+x2)]/[( 1+x2)( 1+x 1)]
=(x2-x 1)/[( 1+x2)( 1+x 1)]
① when x 1 < x2
x2-x 1>0, 1+x2f(x 1)
At this point, the function monotonically increases.
② when-1 < x 1 < x2.
x2-x 1>0, 1+x2>0, 1+x 1>0
∴f(x2)-f(x 1)>0
f(x2)>f(x 1)
At this point, the function monotonically increases.
To sum up, the monotone interval of f(x) is (-∞,-1) and (-1,+∞).
Hope to adopt, O(∩_∩)O Thank you.