Question 1186178
Find the inverse of the function, if it exists. Justify if is does not.

1. {{{p(x)=3x+8}}}

{{{p(x)=y}}}
{{{y=3x+8}}}......swap variables
{{{x=3y+8}}}...solve for {{{y}}}
{{{x-8=3y }}}
{{{y=(x-8)/3}}}

=>{{{p}}}'{{{(x)=(x-8)/3}}}



2. {{{f(x)=x^2+2x - 1}}}

{{{y=x^2+2x - 1}}}.....swap variables
{{{x=y^2+2y - 1}}}
{{{x+1=y^2+2y}}} ...complete square
{{{x+1=(y^2+2y+b^2)-b^2}}}.........b=1
{{{x+1=(y+1)^2-1^2}}}
{{{x+1+1=(y+1)^2 }}}
{{{(y+1)^2=x+2}}}
{{{y+1=srt(x+2)}}}
{{{y=-1}}} ± {{{sqrt(x + 2)}}}
{{{f}}}'{{{(x)=-1}}} ± {{{sqrt(x + 2)}}}



3. {{{g(x)=2x+7}}}

{{{y=2x+7}}}.....swap variables
{{{x=2y+7}}}
{{{2y=x-7}}}
{{{y=(x-7)/2}}}
{{{g}}}'{{{(x)=(x-7)/2}}}



4.{{{f(x)=x^3 - 5}}}

{{{y=x^3 -5}}}...swap variables
{{{x=y^3 - 5}}}
{{{x+5=y^3}}}
{{{y=root(3,x+5)}}}
{{{f}}}'{{{(x)=root(3,x+5)}}}



5.{{{h(x)=(x+2)/(2x-3)}}}

{{{y=(x+2)/(2x-3)}}}...swap variables
{{{x=(y+2)/(2y - 3)}}}
{{{x(2y - 3)=(y+2)}}}
{{{2xy - 3x=y+2}}}
{{{2xy - y=3x+2}}}
{{{(2x - 1)y=3x+2}}}
{{{y=(3x+2)/(2x - 1)}}}

{{{h}}}'{{{(x)=(3x+2)/(2x - 1)}}}