Question

If $g\left(x\right)=x^2+x-1$ and $\left(gof\right)\left(x\right)=4x^2-10x+5,$ 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

Answer: (B)

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