Question 912101
the inverse of {{{f(x) = 4-ln(x+2)}}}


 remember that {{{f(x) =y}}}


{{{y = 4-ln(x+2) }}} ...swap {{{x}}} and {{{y}}} to find the inverse:

{{{x= 4-ln(y+2)}}} ....solve for {{{y}}}

{{{ln(y+2)= 4-x}}}


by definition: {{{log(a ,x )= N}}} means that {{{a^N= x}}}


in {{{ln(y+2)}}} ,you have that {{{a=e}}}, {{{x=y+2}}}, and {{{N=4-x}}}


so, {{{e^(4-x)= y+2}}}


{{{ y=e^(4-x)-2}}}

since {{{f^(-1)(x)=y}}}, we have 

{{{f^(-1) (x) =e^(4-x)-2}}}
or

{{{f^(-1) (x) = -e^(-x) (2e^x-e^4)}}}

domain: {{{R}}}  (all real numbers)

 {{{ graph( 600, 600, -10, 10, -10, 10, 4-ln(x+2), e^(4-x)-2,x) }}}