Question 997925

{{{f(x)= (1/2)x + 1}}}

a. Find the inverse of {{{f(x)}}} and name it {{{g(x)}}}. Show and explain your work.
note: {{{f(x) =y}}}
{{{y= (1/2)x + 1}}}.....to find inverse, swap {{{x}}} and {{{y}}}

{{{x= (1/2)y + 1}}}....solve for {{{y}}}

{{{x-1= (1/2)y }}}

{{{y=(x-1)/(1/2)}}}

{{{y=2(x-1)}}}

{{{y=2x-2}}}-> inverse  {{{g(x)}}}

so {{{ g(x)=2x-2}}}


b. Use function composition to show that {{{f(x)}}} and {{{g(x)}}} are inverses of each other.
if  {{{f(x)}}} and {{{g(x)}}} are inverses of each other, than {{{(f o g)(x) }}} will end up with just "{{{x}}}", so we will have:

{{{(f o g)(x)=x}}}

let's check it:

{{{(f o g)(x)=f(g(x))}}}

{{{(f o g)(x)=f(2x-2)}}}

{{{(f o g)(x)=(1/2)(2x-2) + 1}}}

{{{(f o g)(x)=2x/2-2/2+ 1}}}

{{{(f o g)(x)=x-1+ 1}}}

{{{(f o g)(x)=x}}}=> proof that {{{f(x)}}} and {{{g(x)}}} are inverses of each other