Question

If $f\left(x\right)=3x-2$ and $(gof{)}^{-1}=x-2$, then $\int g\left(x\right)dx$ is (where $C$ is constant of integration)

• A. $\frac{3x^2}{2}-2x+C$
• B. $-\frac{x^2}{6}-\frac{8}{3}x+C$
• C. $\frac{x^2}{6}+\frac{8}{3}x+C$
• D. $\frac{x^2}{2}+x+C$

Answer 1

The correct option is C $\frac{x^2}{6}+\frac{8}{3}x+C$

$f\left(x\right)=3x-2$
Let
$y=3x-2\Rightarrow y+2=3x\Rightarrow \frac{y+2}{3}=x$
$\Rightarrow f^{-1}\left(x\right)=\frac{x+2}{3}\Rightarrow (gof{)}^{-1}=x-2\Rightarrow f^{-1}(g^{-1}\left(x\right))=x-2\Rightarrow f^{-1}(g^{-1}\left(x\right))=x-2\Rightarrow \frac{g^{-1}\left(x\right)+2}{3}=x-2\Rightarrow g^{-1}(x)=3x-8\Rightarrow \frac{g^{-1}\left(x\right)+8}{3}=x\Rightarrow g(x)=\frac{x+8}{3}\Rightarrow \int g(x)dx=\frac{x^2}{6}+\frac{8}{3}x+C$