Composite Functions and Their Domains

The video tutorial explains how to find composite functions from piecewise defined functions using graphical and algebraic methods. It covers three examples: finding f(g(x)) and g(f(x)) using both methods, and finding f(f(x)) and g(f(x)) with detailed steps and explanations. The tutorial emphasizes the importance of understanding function definitions and intervals to accurately determine composite functions.

What are the two methods to find composite functions from piecewise defined functions?

• Algebraic and Numerical

• Numerical and Graphical

• Graphical and Algebraic

• Graphical and Statistical

In the first example, what is the function f(x) defined as when x is between -1 and 1?

• 2 - x

• x - 2

• -x

• x + 2

What is the function g(x) defined as when x is between -2 and -1?

• x + 3

• 1 - x

• 2 - x

• x - 3

Using the graphical method, what is the first step in finding f(g(x))?

• Draw the graph of f(x)

• Find the domain

• Solve algebraically

• Draw the graph of g(x)

In the algebraic method, what is f(g(x)) when g(x) is x + 3 and lies between -1 and 1?

• -x - 1

• x + 3

• x - 1

• x + 1

What is the domain of f(f(x)) in the second example?

• [-2, 2]

• [-4, 4]

• [-4, √6]

• [0, 4]

In the second example, what is the function f(x) when x is between 0 and 4?

• 2 - x^2

• 2 + x^2

• x + 2

• x - 2

Using the graphical method, what is the function g(f(x)) when x is between -1 and 1?

• 1 + x

• 3 - x

• 2x

• x^2

In the third example, what is the function f(x) when x is greater than or equal to 0?

• 2 - x

• x + 1

• 1 - x

• x^2

Using the algebraic method, what is g(f(x)) when f(x) is x^2 and lies between -1 and 1?

• 1 - x^2

• 3 - x^2

• 2x^2

• x + 1