Lecture Notes on Function Notation

Summary of the Lecture

Today's lecture discussed the concept of function notation. We explored how to represent functions in a standard mathematical form, substituting "y" with "f(x)" to indicate that 'f' is a function of 'x'. This was demonstrated using practical examples which showed how to apply this notation to solve for specific values and to combine functions.

Detailed Notes

Definition of Function Notation

• A typical function that you might see can be written as y = 2x + 5.
• Using function notation, this is rewritten as f(x) = 2x + 5.
  • Here, “f” indicates a function, and “x” is the variable.
  • The “y” is replaced by f(x).

Purpose of Using Function Notation

• This notation helps in specifying which value of x is being used.
• Example: f(4) will substitute x = 4 in the function, so f(4) = 2*4 + 5 = 13.
  • This means when x is 4 in this function, the output is 13.*

Application Examples

Example 1

• Given functions f(x) = -5x + 2 and g(x) = 3x - 20.
• Task: Calculate f(4) and g(4).
  • For f(4):
    • Replace x with 4 in f(x), resulting in f(4) = -5*4 + 2 = -18.
  • For g(4):
    • Replace x with 4 in g(x), resulting in g(4) = 3*4 - 20 = -8.

Example 2

• New function h(x) is defined as the sum of f(x) and g(x): h(x) = f(x) + g(x).
• Task: Derive h(x) and simplify.
  • Substitute the expressions for f(x) and g(x):
    • h(x) = (-5x + 2) + (3x - 20).
  • Combine like terms:
    • Combine the x terms: -5x + 3x = -2x.
    • Combine the constant terms: 2 - 20 = -18.
    • Thus, h(x) = -2x - 18.

These examples illustrate how using function notation can simplify the handling and manipulation of multiple functions, particularly useful for solving for specific values and for operations involving multiple functions.