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  4. If g(x)=x2+x-1 and (g o f)(x)=4x2-10x+5, then f((5/4)) is equal to

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Q. If $g\left(x\right)=x^{2}+x-1$ and $\left(g o f\right)\left(x\right)=4x^{2}-10x+5,$ then $f\left(\frac{5}{4}\right)$ is equal to

NTA AbhyasNTA Abhyas 2020

Solution:

$g(f(x))=4\left(\frac{5}{4}\right)^{2}-10 \cdot \frac{5}{4}+5=-\frac{5}{4}$
$g\left(f\left(\frac{5}{4}\right)\right)=f^{2}\left(\frac{5}{4}\right)+f\left(\frac{5}{4}\right)-1$
$-\frac{5}{4}=f^{2}\left(\frac{5}{4}\right)+f\left(\frac{5}{4}\right)-1$
$f^{2}\left(\frac{5}{4}\right)+f\left(\frac{5}{4}\right)+\frac{1}{4}=0$
$\left(f\left(\frac{5}{4}\right)+\frac{1}{2}\right)^{2}=0$
$f\left(\frac{5}{4}\right)=-\frac{1}{2}$