Lessons

Composition of Functions

• Composition involves combining two functions, denoted as f(g(x)) or g(f(x)).

• Example: Given functions

  • f(x) = 2x + 3

  • g(x) = x^2

• Composite Functions:

  • f(g(x)) = 2(x^2) + 3

  • g(f(x)) = (2x + 3)^2

  • Calculate:

    • f(g(x)) = 2(x^2) + 3

    • g(f(x)) = (2x + 3)^2 - 1

      • Expand:

        • = 4x^2 + 12x + 8

        • Final result: g(f(x)) = (x + 1)(x + 2).

Examples of Function Composition

• Given functions:

  • f(x) = 3x - 3

  • g(x) = x - x

• Evaluate:

  • f(g(3))

  • f(g(x)) = 3(x^2 - x) - 5

    • Result: 3(6) - 31 = 3x^2 - 3x - j

    • Found: 18 - 5 = 13

• Final composite for given functions:

  • g(f(x)) = (3x - 5)^2 - (3x - 5).

Inverse Functions

• Concepts:

  • The inverse function denoted as f^{-1}(x) satisfies:

    • f(f^{-1}(x)) = x

  • Example inverse calculation for h(x) = x^3

    • Functions such that:

      • If f(x) = x^3 + 1, find f^{-1}(x) where y = x^3, leading to f(g(x)) = x.

Average Rate of Change

• The average rate of change is determined by the slope of the secant line connecting two points on the graph of a function.

• Example:

  • For function y = x:

    • Points chosen: (1,1) and (4,3)

    • Find slope, m.

Examples of Average Rate

• Method:

  • Choose two points (x1,y1) and (x2,y2) from the function.

  • Slope calculation:

    • m = (y2 - y1) / (x2 - x1)

  • Average slope identified for intervals defined in examples.

Instantaneous Rate of Change

• As the interval narrows, the secant line approaches the tangent line, which indicates the instantaneous rate of change.

• Example Functions:

  • For f(x) = x^2, derivative approaches instantaneous slope at a given x.

Applications of Rates of Change

• Real-world scenarios employing rates of change calculations, e.g., average velocity of thrown objects, profit calculations in business.

• Base Applicative Example: M(t) = -5t^2 + 20t + 2

  • Where:

    • a) Average velocity from time t=0 to t=5.

    • b) Instantaneous velocity at t=3.