Math 112 – Section 2.8: Algebra of Functions and Function Composition

Math 112 – Section 2.8 Notes: Algebra of Functions and Function Composition

1. Perform Operations on Functions

Example 1: Evaluate the Functions for Given Values of x

  • Functions A, B, and C are to be evaluated based on values of x provided in examples.

Definition of Function Operations

Combining functions with the basic operations (sum, difference, product, and quotient) to result in a function that can be evaluated, graphed, and analyzed is the algebra of functions.

Algebra of Functions

  • Given functions f and g, the operations are defined as follows:
    • Sum of Functions:
      (f + g)(x) = f(x) + g(x)
    • Difference of Functions:
      (f - g)(x) = f(x) - g(x)
    • Product of Functions:
      (f imes g)(x) = f(x) imes g(x)
    • Quotient of Functions:
      (f / g)(x) = \frac{f(x)}{g(x)}
  • Note: The domain of the functions f and g is all real numbers in the intersection of the domains of the individual functions f and g. For the quotient \frac{f}{g} , we further restrict the domain to exclude values of x for which g(x) = 0 .

Given Functions

  • f(x) = -\frac{1}{2}x^2 + 3
  • g(x) = -8
  • h(x) = 2

Operations to Evaluate:

  • (f + g)(4)
  • (g - h)(7)
  • (f \cdot h)(2)

2. Evaluate a Difference Quotient

Definition of Average Rate of Change

The average rate of change between points P and Q is the slope of the secant line and is given by:
m = \frac{f(Q) - f(P)}{Q - P}
This expression is called the difference quotient.

Given Functions

  • f(x) = x^2 - 9
  • g(x) = 4x^3 - 12
  • h(x) = -x^4

Operations to Evaluate:

  • (g + f)(x)
  • (f - g)(x)
  • (f \cdot h)(x)
  • (h / g)(x)

3. Compose and Decompose Functions

Definition and Notation of Function Composition

When the input value is itself a function, this process is called the composition of functions.

  • The notation used for the composition of f with g is given as (f \circ g) , which uses an open dot notation and should not be confused with the multiplication dot used to indicate the product of two functions.

Example Functions

  • f(x) = -x + 5
  • g(x) = x^2 + 2

Composition to Evaluate:

  • Evaluate f(g(x))
  • Evaluate g(f(x))

4. Definition of Composition of Functions

The composition of f and g, denoted as (f \circ g) , is defined by:
(f \circ g)(x) = f(g(x))

  • The domain of f \circ g is the set of real numbers x in the domain of g such that g(x) is in the domain of f. Similarly, this also applies for (g \circ f) .

Example Functions

  • f(x) = 3x - 4
  • g(x) = x^2 + 8

Evaluation of Compositions

  • (f \circ g)(0)
  • (g \circ f)(2)
  • (h \circ f)(12)

5. Combining Concepts

Example 9: Evaluate Function with Graphs

Given the graphs of two functions, evaluate at specified x values, if possible.

  • Possible evaluations:
    • (f - g)(3)
    • (g + f)(0)
    • (f \cdot g)(1)
    • (f / g)(-2)
    • (g - f)(3)
    • Additional evaluations at (f)(x) and (g)(x) for multiple x values.

Note

You should be able to find the sum, difference, product, quotient, and composition of functions from a graph, indicating a comprehensive understanding of function operations and their applications.

Additional Example Functions and Evaluations

Example 7: Find Indicated Function

  • Identify functions A, B, C, and D and determine their compositions.

Note on Composite Functions

The above operations involving combinations of functions is called creating composite functions.

Example Functions Recap

  • Given functions:
    • f(x) = 2x - 11
    • g(x) = 3x + 5
    • h(x) = 5x + 5
    • j(x) = x - 1
  • Evaluate combinations such as (h \circ g)(x) and (j \circ f)(x) to solidify understanding of function composition.