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