Question

For what value of x is $fog=gof$ if $f\left(x\right)=x-2$ and $g\left(x\right)=x^3+3$?

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

Answer 1

The correct option is B $-1$

$f\left(x\right)=x-2g\left(x\right)=x^3+3f\left[g\right(x\left)\right]=g\left(x\right)-2\Rightarrow fog=x^3+3-2=x^3+1g\left[f\right(x\left)\right]={\left\{f\right(x\left)\right\}}^3+3\Rightarrow gof={(x-2)}^3+3fog=gof{x}^3+1={(x-2)}^3+3x^3-2={(x-2)}^3{x}^3-2=x^3-8-6x(x-2)x^3-2=x^3-8-6x^2+12x\Rightarrow x^2+1+2x=0\Rightarrow {(x+1)}^2=0\Rightarrow x=-1$

So none of the above options are correct