Find the specified function. (g ? f)-1

Question

Use the functions
f(x) = x + 3
and
g(x) = 2x − 1
to find the specified function.

 

(g o f)<sup>-1

(g o f)-1(x) = ?</sup>

Answer 1

(gof) (x) = g(f(x))

g(f(x)) = g(x+3) = 2(x+3)-1 =2x+5

 

(g o f)^-1(x) =g(f(x))^-1 

 

let's denote g(f(x)) as y: 

y = 2x+5

 

To find the inverse of the function, switch the x and y:

x = 2y+5

 

solve for y:

x-5= 2y

x/2 -5/2 = 2y/2

 

y = x/2 -5/2

 

this is the (g o f)-¹(x) .

Answer 2

f(x) = x+3   g(x) = 2x-1

 

(gºf)(x) = g(f(x)) = g(x+3) = 2(x+3) - 1 = 2x + 5

 

To find the inverse function for (gºf)(x):

 

   Step 1 Set y = 2x+5

   Step 2 Switch x and y to get x = 2y+5

   Step 3 Solve for y.  The result is the inverse function.

 

                  2y = x-5

 

                    y = (x-5)/2

 

   So, (gºf)^-1(x) = (x-5)/2