Question

If F:R>R and g:R>R defined by f(x)=2x+1 and g(x)=2^x find the equation for (gof) and (fog)?

Also find f^(-1)(x) and g(3)?

Answer 1

`gof(x)= 2^(2x+1)`
`fog(x) =2*2^x+1`
`f^-1(x)=(x-1)/2`
`g(3)=8`

Explanation:

It should be cleared that,
`gof(x)= g(f(x))` and`fog(x) =f(g(x))`
Given-
`f(x)=2x+1` and `g(x)=2^x`

1. `gof(x):`
   `gof(x)= g(f(x))`
   `:.` `gof(x)= g(2x+1)`
   `:.` `gof(x)= 2^(2x+1)`
2. `fog(x) :`
   `:.` `fog(x) =f(2^x)`
   `:.` `fog(x) =2*2^x+1`
3. `f^-1(x):`
   Let , `y= 2x+1`
   `=>` `x=(y-1)/2`
   Hence, `f^-1(x)=(x-1)/2`
4. `g(3):`
   `g(x)=2^x`
   `g(3)=2^3`
   `g(3)=8`