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[Solved] Functions equations

 
Quadratic Equations
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June 15, 2021 11:26 am
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(@armor)
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If the equation   (2-2^{-|x-3|})^2=3+a  with respect to x has real roots, find the range of real number a.

 

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June 15, 2021 11:26 am
Topic starter
Staff 
(@armor)
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317 Posts
156 153 0

We simply the given equation to get   \small a=(2-2^{-|x-3|})^2-3.

Let \small t=2^{-|x-3|},  then  \small 0<t\leq 1,   \small a=f(t)=(t-2)^2-3.

Since  \small a=f(t) is decreasing on \small (0,1],  then \small f(1)\leq f(t)<f(0).

This means \small -2\leq f(t)<1.  Thus the range of real number \small a is \small a\in [-2,1).

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