18. If \( 2^{f(x)}=\frac{2+x}{2-x}, x \in(-2,2) \) and \( f(x)=\lambda f\left(\frac{8 x}{4+x^{2}}\right) \) then value of ' \( \lambda \) ' will be (1) 2 (2) \( \frac{1}{2} \) (3) 1 (4) -1

Question

18. If \( 2^{f(x)}=\frac{2+x}{2-x}, x \in(-2,2) \) and \( f(x)=\lambda f\left(\frac{8 x}{4+x^{2}}\right) \) then value of ' \( \lambda \) ' will be (1) 2 (2) \( \frac{1}{2} \) (3) 1 (4) -1

Answer 1

2^f(x) = \(\frac{2+x}{2-x}\)   ......(1)

f(x) = \(\lambda f(\frac{8x}{4+x^2})\)

f(1) = \(\lambda f(\frac{8}{5})\)  ......(2)

2^f(1) = \(\frac31=3\)    From (i)

⇒ f(1) = log₂3

2^{f(\(\frac85\))} = \(\frac{2+\frac85}{2-\frac85}\)

= \(\frac{18}2 =9\) 

\(\therefore\) f(\(\frac85\)) = log₂9 = log₂3² = 2log₂³

\(\therefore\) 2\(\lambda\) log₂3 = log₂3  From (2)

f\(\lambda\) = \(\frac12\)