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

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(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.