Question

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

• A. $-\frac{3}{2}$
• B. $-\frac{1}{2}$
• C. $\frac{1}{2}$
• D. $\frac{3}{2}$

Answer 1

The correct option is B $-\frac{1}{2}$

$g\left(x\right)=x^2+x-1$
$(g\circ f)\left(x\right)=4x^2-10x+5$
$\Rightarrow g\left(f\right(x\left)\right)=4x^2-10x+5$
$\Rightarrow f^2\left(x\right)+f\left(x\right)-1=4x^2-10x+5$
Putting $x=\frac{5}{4}$ and $f\left(\frac{5}{4}\right)=t$
$t^2+t+\frac{1}{4}=0$
$\Rightarrow t=-\frac{1}{2}$ or $f\left(\frac{5}{4}\right)=-\frac{1}{2}$