Find an inverse function of the function if it exists Fx squ

Find an inverse function of the function if it exists. F(x)= squareroot x+2 g(x)=-3/4x-3 h(x)=2x/x+7 Find (f g)(x) where f(x)=2x^2-x+1,g(x)=g(x)=x+1. Simplify your result.

Solution

1. f(x) = (x+2)^1/3

Inverse function : A function says that for every x, there is exactly one y. That is, y values can be duplicated but x values can not be repeated

y = (x+2)^1/3

To find inverse plug x=y and y= x and solve for y:

x = (y+2)^1/3

y = x^3 -2

f^-1(x) =  x^3 -2

2. g(x) = -3/( 4x -3)

y =-3/( 4x -3)

inverse for this function exists

To find inverse plug x=y and y= x and solve for y:

x = -3/(4y -3)

4xy -3x = -3

4xy = 3x -3

y = (3x -3)/4x

f^-1(x) =(3x -3)/4x

3. h(x) = 2x/(x+7)

inverse for this function exists.

To find inverse plug x=y and y= x and solve for y:

y = 2x/(x+7)

x = 2y/(y+7)

xy +7x = 2y

y( x -2) = -7x

y = -7x/(x-2) = 7x/(2-x)

f^-1(x) =7x/(2-x)

4. find ( fog)(x)

f(x) = 2x^2 -x +1 ; g(x) = x+1

plug x = g(x) in f(x)

( f og)(x) = 2(x+1)^2 -(x+1) +1

= 2(x^2 +1 +2x) -2x -2 +1

= 2x^2 +2 +4x -2x -1

= 2x^2 +2x +1