Composition of Functions

Study Notes

Composition of Functions

Function Definitions:
- ( f(x) = 4x + 3 )
- ( g(x) = x - 2 )
Example Calculation:
- For ( f(g(5)) ):
  - Calculate ( g(5) = 5 - 2 = 3 )
  - Then ( f(3) = 4(3) + 3 = 15 )
Example Calculation:
- For ( g(f(-6)) ):
  - Find ( f(-6) = 4(-6) + 3 = -21 )
  - Then ( g(-21) = -21 - 2 = -23 )
Example Calculation:
- For ( f(f(7)) ):
  - Calculate ( f(7) = 4(7) + 3 = 31 )
  - Then ( f(31) = 4(31) + 3 = 127 )
Composite Function:
- For ( g(f(x)) ):
  - Start with ( f(x) = 4x + 3 )
  - Then calculate ( g(4x + 3) = (4x + 3) - 2 = 4x + 1 )

Alternative Functions

Using Different Functions:
- If ( f(x) = 6x^2 ) and ( g(x) = 14x + 4 ):
  - Composition ( (f \circ g)(x) ):
    - Calculate ( g(x) = 14x + 4 )
    - Then ( f(g(x)) = f(14x + 4) = 6(14x + 4)^2 = 1176x^2 + 672x + 48 )
Other Composition:
- For ( (g \circ f)(x) ):
  - Start with ( f(x) = 6x^2 )
  - Then calculate ( g(f(x)) = g(6x^2) = 14(6x^2) + 4 = 84x^2 + 4 )

Key Observations

Difference in Results:
- The results of ( (f \circ g)(x) ) and ( (g \circ f)(x) ) are not the same.
- This demonstrates that function composition is not commutative; the order matters.
- Understanding composition typically requires familiarity with the specific functions involved.

Description

Test your understanding of function composition with this quiz on f(x) and g(x). The questions involve evaluating combined functions and their impacts. Perfect for students looking to solidify their knowledge of function operations.