Let f(x)= array|c|c| ax^2+2ax+34x^2+4x-3 & ,x -(3)/(2),(1)/(2) b & ,x=-(3)/(2),(1)/(2) arr…

Question

Let $f(x)= \begin{array}{|c|c|} \frac{a{x}^{2}+2ax+3}{4{x}^{2}+4x-3} & ,x
e -\frac{3}{2},\frac{1}{2} \ b & ,x=-\frac{3}{2},\frac{1}{2} \end{array}$ be continuous at $x=-\frac{3}{2}$. If $f\circ f(x)=\frac{7}{5}$, then $x$ is equal to:

Accepted Answer

$\begin{gathered} \underset{x	o \frac{-3}{2}}{lim}\frac{a{x}^{2}+2ax+3}{4{x}^{2}+4x-3}=b \ a{x}^{2}+2ax+3=0\, n=\frac{-3}{2} \ \Rightarrow \frac{9a}{4}-3a+3=0 \ \Rightarrow \frac{3a}{4}=3 \ \Rightarrow a=4 \ \underset{x	o \frac{-3}{2}}{lim}\frac{4{x}^{2}+8ax+3}{4{x}^{2}+4x-3} \end{gathered}$

$\begin{gathered} =\underset{x	o \frac{-3}{2}}{lim}\frac{(2x+1)(2x+3)}{(2x-1)(2x+3)} \ =\frac{-2}{-4}=\frac{1}{2}=b \ 	herefore f(x)=\frac{2x+1}{2x-1} \ f(f(x))=\frac{7}{5} \ 	herefore \, \frac{2f(x)+1}{2f(x)-1}=\frac{7}{5} \ \Rightarrow f(x)=3 \ \frac{2x+1}{2x-1}=3 \ \Rightarrow x=1 \end{gathered}$

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