# Let f:$R\rightarrow R$ be the functions defined by $f(x)=x^3+5.$ Then $f^{-1}(x)$ is

\begin{array}{1 1}(A)\;(x+5)^{\frac{1}{3}} & (B)\;(x-5)^{\frac{1}{3}}\\(C)\;(5-x)^{\frac{1}{3}} & (D)\;5-x\end{array}

Toolbox:
• A function g is inverse of $f:R \to R$ if $fog=gof=I_R\;ie \;g=f^{-1}$
$f:R \to R$

$f(x)=x^3+5$

Let $y=x^3+5 \qquad y \to R$

$x^3=y-5$

$x=(y-5)^{1/3}$

we define a function $g(y)=(y-5)^{1/3}$

$(gof)(x)=g(f(x))$

$=g(x^3+5)$

$=(x^3+5-5)^{1/3}=(x^3)^{1/3}=x$

Also $fog(g)=f(g(y))$

$=f\bigg((y-5)^{1/5}\bigg)$

$=\bigg[(y-5)^{1/5}\bigg]^3+5$

$=y-5+5$

=y

=>$fog=gof=I_R$

$f^{-1}(x)=g(x)=(x-5)^{1/3}$

'B' option is correct