Function Operations Overview

Function Operations Overview

• Function operations involve combining two or more functions through addition, subtraction, multiplication, and division.

Function Addition

• Defined as: ( (f + g)(x) = f(x) + g(x) )
• The sum of two functions produces a new function.
• Example:
  • Let ( f(x) = 2x + 3 ) and ( g(x) = x^2 ).
  • Then, ( (f + g)(x) = (2x + 3) + (x^2) = x^2 + 2x + 3 ).

Function Subtraction

• Defined as: ( (f - g)(x) = f(x) - g(x) )
• The difference of two functions gives a new function.
• Example:
  • Let ( f(x) = 3x + 5 ) and ( g(x) = x^2 ).
  • Then, ( (f - g)(x) = (3x + 5) - (x^2) = -x^2 + 3x + 5 ).

Function Multiplication

• Defined as: ( (f \cdot g)(x) = f(x) \times g(x) )
• The product of two functions results in a new function.
• Example:
  • Let ( f(x) = x + 1 ) and ( g(x) = 2x ).
  • Then, ( (f \cdot g)(x) = (x + 1)(2x) = 2x^2 + 2x ).

Function Division

• Defined as: ( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} ) (where ( g(x) \neq 0 ))
• The quotient of two functions results in a new function, subject to the domain restriction where ( g(x) \neq 0 ).
• Example:
  • Let ( f(x) = x^2 + 1 ) and ( g(x) = x - 2 ).
  • Then, ( \left(\frac{f}{g}\right)(x) = \frac{x^2 + 1}{x - 2} ).

Composition of Functions

• Defined as: ( (f \circ g)(x) = f(g(x)) )
• Composition allows one function to be applied inside another.
• Example:
  • Let ( f(x) = x + 1 ) and ( g(x) = 2x ).
  • Then, ( (f \circ g)(x) = f(g(x)) = f(2x) = 2x + 1 ).

Key Points

• Each operation produces a new function based on the inputs from the original functions.
• Understanding the domain is crucial, especially in division and composition, to avoid undefined expressions.
• Function operations can be graphically represented and analyzed for insights into their behavior and intersections.

Function Operations Overview

• Function operations unify multiple functions through addition, subtraction, multiplication, and division to create new functions.

Function Addition

• Addition of functions follows the rule: ( (f + g)(x) = f(x) + g(x) ).
• Example:
  • For ( f(x) = 2x + 3 ) and ( g(x) = x^2 ), the resulting function is ( (f + g)(x) = x^2 + 2x + 3 ).

Function Subtraction

• Subtraction of functions is expressed as: ( (f - g)(x) = f(x) - g(x) ).
• Example:
  • With ( f(x) = 3x + 5 ) and ( g(x) = x^2 ), the outcome is ( (f - g)(x) = -x^2 + 3x + 5 ).

Function Multiplication

• Multiplication operates under the definition: ( (f \cdot g)(x) = f(x) \times g(x) ).
• Example:
  • If ( f(x) = x + 1 ) and ( g(x) = 2x ), then ( (f \cdot g)(x) = 2x^2 + 2x ).

Function Division

• Division of functions is formulated as: ( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} ) with the condition ( g(x) \neq 0 ).
• Example:
  • For ( f(x) = x^2 + 1 ) and ( g(x) = x - 2 ), the division yields ( \left(\frac{f}{g}\right)(x) = \frac{x^2 + 1}{x - 2} ).

Composition of Functions

• Composition is given by: ( (f \circ g)(x) = f(g(x)) ).
• Example:
  • With ( f(x) = x + 1 ) and ( g(x) = 2x ), the composition results in ( (f \circ g)(x) = 2x + 1 ).

Key Points

• Each operation creates a new function influenced by the initial functions' outputs.
• Awareness of domain restrictions is essential, particularly in division and composition, to prevent undefined results.
• Function behaviors, relationships, and intersections can be visualized through graphs, providing insightful analyses.