MTH1120: Combining Functions and Composite Functions, 4.2

MTH1120 Notes on Combining Functions and Composite Functions

Example 1: Operations on Functions

  • Functions Defined:
    • f(x) = x² - 1
    • g(x) = 7 + x
  • Operations:
    • Perform addition, subtraction, multiplication, or division of f and g.
    • Domain considerations: Ensure that the operations are valid based on the functions’ definitions.

Example 2: Revenue, Cost, and Profit Calculations

  • Demand Function:
    • p(x) = 1000 - 2x (Price per unit based on quantity x)
  • Cost:
    • Fixed Cost: $1999
    • Variable Cost: $4 per unit
    • Total Cost Function:
      • C(x) = 4x + 1999
  • Revenue Function:
    • R(x) = 1000x - 2x²
  • Profit Function:
    • To find profit:
      • Profit P(x) = Revenue - Cost
      • So, P(x) = R(x) - C(x) = (1000x - 2x²) - (4x + 1999)
      • Simplified Profit Function:
      • P(x) = -2x² + 996x - 1999
  • Maximizing Profit:
    • To find the number of units x for maximum profit, find the vertex of the parabola defined by the profit function:
      • Vertex formula: x = -b/(2a) where a and b are coefficients from P(x).
      • Here, a = -2, b = 996
      • Calculation:
      • x = -996 / (2 * -2) = 124.5, round appropriately as units sold must be a whole number (e.g., 124 or 125 units).
    • Maximum Profit Calculation:
      • Substitute x back into P(x) to find the maximum profit value.

Example 3: Finding Composite Function Outputs

  • Functions Given:
    • b(x) = √x - 1
    • f(x) = 2x - 5
    • g(x) = 6 - x²
    • h(x) = 1/x
    • m(x) = (x + 3)/5
  • Composite Functions:
    • Perform composition of functions where applicable (e.g., b(f(x)), f(g(x)), etc.).
    • Ensure to state the domain of each new function created based on the original functions.

Example 4: Evaluation Using Function Graphs

  • Evaluate Expressions:
    • (f + g)(-1): Sum of outputs of f and g at x = -1.
    • (f∙g)(5): Product of outputs of f and g at x = 5.
    • (gf)(3): Output of g applied to the output of f at x = 3.

Composition Practice

  • Objective: Create new composite functions from provided functions.
  • Example Compositions:
    • For example, forming h(j(x))
    • Functions provided:
      • (M = 3√{17 - 5x});
      • (P = 1/(17-5x));
      • (Q = 1/(6x⁴));
      • (R = 17 - 5√x);
      • (T = x);
      • Additional functions given:
      • f(x) = √x, g(x) = 1/x, h(x) = 3√x, j(x) = x², k(x) = 17 - 5x, n(x) = 1/(6x²).
  • Goal: Combine at least two of the provided functions to generate new expressions while maintaining functional integrity.