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2^x- 1/2^|x|=2

x & gt0

2^x- 1/2^x=2

2^x= 1+√2

x=log2( 1+√2)

x & lt0

2^x- 1/2^-x=2

2 x-2 x = 2 is invalid.

g(t)=2^tf(2t)+mf(t)

=2^t[2^(2t)- 1/2^(2t)]+m(2^t- 1/2^t)

=2^(3t)- 1/2^t+m(2^t- 1/2^t)

x=2^t

2 & lt= x & lt=4

4 & lt=x^2<; = 16

g(x)=x^3- 1/x+m(x- 1/x)=(x^4+mx^2- 1-m)/x>; =0

x^4+mx^2- 1-m>; =0

x & gt0

x^2>; =-m/2+√(m^2+4+4m)/2=-m/2+|m+2|/2

x^2<; =-m/2-√(m^2+4+4m)/2=-m/2-|m+2|/2

m & lt-2

x^2>; =-m/2+√(m^2+4+4m)/2=-m/2-m/2- 1=-m- 1<; 4 does not hold water

x^2<; =-m/2-√(m^2+4+4m)/2=-m/2+m/2+ 1= 1<; 4 does not hold water

m & gt=-2

x^2>; =-m/2+√ (m 2+4+4m)/2 =-m/2+m/2+1=1hold.

x^2<; =-m/2-√(m^2+4+4m)/2=-m/2-m/2- 1=-m- 1<; = 16

-m- 1 & lt; = 16

m & gt=-2