Question

Find the inverse of the function. f(x) = the cube root of quantity x divided by seven. - 9

Answer 1

To solve

replace f(x) with y
switch x and y
solve for y
replace y with f⁻¹(x)

so
I'm not sure if it is
$f(x)=\sqrt[3]{\frac{x}{7}}-9$ or
$f(x)=\sqrt[3]{\frac{x}{7}-9}$


first one
$f(x)=\sqrt[3]{\frac{x}{7}}-9$
replace f(x) with y
$y=\sqrt[3]{\frac{x}{7}}-9$
switch x and y
$x=\sqrt[3]{\frac{y}{7}}-9$
solve for y
$x+9=\sqrt[3]{\frac{y}{7}}$
$(x+9)^3=\frac{y}{7}$
$7(x+9)^3=y$
replace y  with f⁻¹(x)
$f^{-1}(x)=7(x+9)^3$

2nd one
$f(x)=\sqrt[3]{\frac{x}{7}-9}$
replce f(x) with y
$y=\sqrt[3]{\frac{x}{7}-9}$
switch x and y
$x=\sqrt[3]{\frac{y}{7}-9}$
solve for y
$x^3=\frac{y}{7}-9$
$x^3+9=\frac{y}{7}$
$7(x^3+9)=y$
replace y  with f⁻¹(x)
$f^{-1}(x)=7(x^3+9)$



if you meant $f(x)=\sqrt[3]{\frac{x}{7}}-9$  then the inverse is $f^{-1}(x)=7(x+9)^3$

if you meant $f(x)=\sqrt[3]{\frac{x}{7}-9}$  then the inverse is $f^{-1}(x)=7(x^3+9)$