Prove monotonicity of function by definition
Specific steps:
(1) takes any two values.
(2) Doing errands
(3) Deformation (factorization, formula, physicalization, etc. )
(4) number
(5) draw a conclusion
Prove:
Let x 1, x2 ∈ (- 1, +∞), x 1 < x2.
Then f (x 1)-f (x2)
=a^(x 1)+[(x 1-2)/(x 1+ 1)]-a^(x2)+[(x2-2)/(x2+ 1)]
=[a^(x 1)-a^(x2)]+[[(x 1-2)(x2+ 1)-(x2-2)(x 1+ 1)]/(x 1+ 1)(x2+ 1)]
=[a^(x 1)-a^(x2)]+[ 3(x 1-x2)/(x 1+ 1)(x2+ 1)]
∵ y = a x (a > 1) is the increasing function in the definition.
∴a^x 1
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