Composition of Functions Assignment

Given: f(x)= x^2-3 & g(x)= x+1, what is the composite function g(f(x))?

Given: f(x)= 1/x+5 & g(x)= x-2, what restriction applies?

Given: f(x) = 2x + 5 and g(x) = x^2 and h(x) = -2x, what is h(g(f(x)))?

What is f(x) if f(x)= x^2 +1?

Given: g(x)= sqrt(x-4) & h(x)=2x-8, what are the restrictions on the domain of g(h)?

Given: g(x)= sqrt(x-4) & h(x)=2x-8, what is g(h(10))?

Use the graph to evaluate the composition f(g(0)). What is the result?

Explain how to write a composite function to find the area of the region at time t for the oil spill given r(t) = 0.5 + 2t and a(r) = πr^2.

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).

Test your understanding of the composition of functions with this assignment flashcard set. The quiz covers evaluating composite functions, determining specific values, and understanding function definitions. Perfect for students looking to reinforce their knowledge in algebra.