Operations on functions

-- Section 3.6 --
 

 Composite Functions

(f og)(x) = f(g(x))   The circle means composite.
 (g of)(x) = g(f(x)) 
   

Let   f(x) = 2x - 1 and g(x) = 5x.  Find (f og)(x)and  (g o f)(x) 

Solve

Step

(f og)(x) = f(g(x)) = f(5x)

Substitute g(x) into f(x)

              = 2(5x) - 1 = 10x - 1

We have a new function 10x - 1

(g of)(x) = g(f(x)) = g(2x - 1)

Substitute f(x) into g(x)

              = 5(2x - 1) = 10x - 5

We have a new function 10x - 5
  

Notice that 10x - 1 ¹   10x - 5                                      (f og)(x) and  (g o f)(x)  may not be equal
    

Let   f(x) = 3x + 2 and g(x) = 2x - 4.  Find  (f og)(x) and  (g o f)(x) 

[Solution]

Let   f(x) = 4x2 + 1 and g(x) = x - 1.  Find  (f og)(x) and  (g o f)(x) 

Solve

Step

(f og)(x) = f(g(x)) = f(x - 1)

Substitute g(x) into f(x)

              = 4(x - 1)2 + 1 

Don't forget the "+ 1"

              = 4(x2 - 2x + 1) + 1 

(x - 1)2  = (x - 1)(x - 1) = x2 - x - x + 1 = x2 - 2x + 1

              = 4x2 - 8x + 4 + 1 

We have a new function 10x - 1

           = 4x2 - 8x + 5

  

(g of)(x) = g(f(x)) = g(4x2 + 1)

Substitute f(x) into g(x)

               = (4x2 + 1) - 1 = 4x2

Simplify

               = 4x2

We have a new function 4x2
  

Notice that 4x2 - 8x + 5 ¹ 4x2                                      (f og)(x) and  (g o f)(x)  may not be equal

 

Let   f(x) = 2x2 + 3 and g(x) = x - 3.  Find  (f o g)(x) and  (g o f)(x)  

[Solution]

Let   f(x) = 4x2 + 1 and g(x) = x - 1.  Find (f o g)(2) and  (g o f)(-5) 

Solve

Step

Check

Another way First find 

From Example 2

Find (f o g)(2) = f(g(2)) = ?

First find g(2)
  		(f og)(x) = 4x2 - 8x + 5
g(2) = 2 - 1 = 1 Substitute 1 into f(x)
 		(f ° g)(2) = 4(2)2 - 8(2) + 5

f(1) = 4(1)2 + 1 = 4(1) + 1 = 5

 (f ° g)(1) = 5

 (f ° g)(2) = 16 - 16 + 5 = 5
   

 

Find (g of)(-5) = g(-5) = ?

First find f(-5)
  		(g of)(x)  =  4x2
f(-5) = 4(-5)2 + 1 = 4(25) + 1 = 101 Substitute 101 into g(x)
 (g of)(-5) = 4(-5)2 = 2(25) =100
g(101) = 101 - 1 = 100

Let   f(x) = 2x2 + 3 and g(x) = x - 3.  Find (f o g)(3) and  (g o f)(-4) 

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada