Composition of Functions Assignment

Function Composition Overview

• Function composition involves combining two or more functions to create a new function that expresses one function applied to the result of another.
• Denoted as f(g(x)), where g is applied first, followed by f.

Example Compositions and Values

• For f(g(13)), the result is 32.
• For f(g(0)), the value obtained is 1.

Specific Function Compositions

• Given f(x) = x^2 - 3 and g(x) = x + 1, the composite function g(f(x)) simplifies to x^2 - 2.
• With functions f(x) = 1/(x+5) and g(x) = x - 2, certain restrictions apply: x cannot equal -3.

Higher-Order Compositions

• For functions f(x) = 2x + 5, g(x) = x^2, and h(x) = -2x, the composition h(g(f(x))) results in -8x^2 - 40x - 50.
• When f(x) = x^2 + 1, the composed result is x^4 + 2x^2 + 2.

Domain Restrictions

• For g(x) = √(x - 4) and h(x) = 2x - 8, the domain restriction states that x must be at least 6 due to the square root function.
• Evaluating g(h(10)) gives the result 2√2.

Graphical Evaluation

• f(g(0)) can be determined through graphical analysis, yielding a result of 3.

Real-World Application

• The radius of a circular oil spill can be modeled by r(t) = 0.5 + 2t, with the area of the spill given by a(r) = πr^2.
• The composite function for the area over time can be expressed as a(r(t)), achieved by substituting r(t) into the area formula: a(r(t)) = π((0.5 + 2t)^2).