Question

If $f\left(x\right)=3x^2-5x-1$ and $(f\circ g)\left(x\right)=3x^2+7x+1,$ then which of the following option is $\text{INCORRECT}?$

• A. $g\left(x\right)=x+2$
• B. $g\left(x\right)=-x-\frac{1}{3}$
• C. $g\left(x\right)=-2x-1$
• D. $g^{\prime }\left(0\right)=\pm 1$

Answer 1

The correct option is C $g\left(x\right)=-2x-1$

$(f\circ g)\left(x\right)=3x^2+7x+1$
$f\left(g\right(x\left)\right)=3x^2+7x+1\cdots \left(1\right)$
Let $g\left(x\right)=ax+b$
So, $f\left(g\right(x\left)\right)=3(ax+b{)}^2-5(ax+b)-1\cdots (2)$
Equating eqn $\left(1\right)$ and eqn $\left(2\right),$
$3x^2+7x+1=3a^2{x}^2+(6ab-5a)x+3b^2-5b-1$
Comparing the coefficient
$a=\pm 1,b=2,-\frac{1}{3}$

$\therefore g\left(x\right)=x+2$ and $g\left(x\right)=-x-\frac{1}{3}$