Question

If the function f and g are defined from the set of real numbers R to R such that $f\left(x\right)=e^x,g\left(x\right)=3x-2$ then find functions fog and gof. Also find the domains of the functions $(fog{)}^{-1}$ and $(gof{)}^{-1}$

• A. $(2,\infty )$
• B. $(-2,\infty )$
• C. $(-2,2)$
• D. $(-\infty ,-2)$

Answer 1

Answer: (B)

$(-2,\infty )$

Given, $f\left(x\right)=e^x$, $g\left(x\right)=3x-2$

$fog$ =$f\left(g\right(x\left)\right)$ $=e^{3x-2}$

Now $(fog{)}^{-1}$

Let $y=e^{3x-2}$

$\therefore \ln y=3x-2$

$x=\frac{\ln y+2}{3}$

Replacing x and y, gives us

$(fog{)}^{-1}=\frac{\ln x+2}{3}$

Therefore domain is $x>0$.

Consider $gof$ $=g\left(f\right(x\left)\right)$ $=3e^x-2$

Therefore $y=3e^x-2$

$y+2=3e^x$

$x=\ln \left(\frac{y+2}{3}\right)$

Replacing x and y give us

$(gof{)}^{-1}=\ln (\frac{x+2}{3})$

Thus domain will be

$x+2>0$

$x>-2$

$\Rightarrow xϵ(-2,\infty )$.