Understanding Composite Functions and Their Domains

The video tutorial explains how to find and simplify the composite function f(g(x)) and determine its domain. It begins by introducing the concept of composite functions and the role of g(x) as the input to f(x). The tutorial then demonstrates how to simplify f(g(x)) using square roots and rationalization techniques. Finally, it provides a detailed explanation of how to determine the domain of the composite function by considering domain restrictions of both g(x) and f(g(x)).

What is the initial step in finding f(g(x))?

• Find the integral of g(x)

• Substitute f(x) into g(x)

• Find the derivative of f(x)

• Substitute g(x) into f(x)

How can the square root of a fraction be simplified?

• By multiplying the numerator and denominator

• By taking the square root of the numerator and denominator separately

• By subtracting the numerator from the denominator

• By adding the numerator and denominator

What is the purpose of rationalizing the denominator?

• To make the expression more complex

• To eliminate the square root from the denominator

• To simplify the numerator

• To add more terms to the expression

What is the domain of a composite function f(g(x))?

• The set of inputs x in the domain of g(x) for which g(x) is in the domain of f(x)

• The set of inputs x in the domain of f(x) for which f(x) is in the domain of g(x)

• The set of inputs x in the domain of f(x)

• The set of inputs x in the domain of g(x)

What is the domain restriction for g(x) = 1/(x+1)?

• x cannot equal 0

• x cannot equal 2

• x cannot equal 1

• x cannot equal -1

Why must the radicand be non-negative in the composite function?

• To ensure the function is real-valued

• To ensure the function is undefined

• To ensure the function is complex

• To ensure the function is imaginary

What happens if x equals -1 in the function g(x)?

• The function is defined

• The function is zero

• The function is infinite

• The function is undefined

What inequality represents the domain of the composite function?

• x ≤ -1

• x < -1

• x > -1

• x = -1

How is the domain of f(g(x)) expressed in interval notation?

• (-∞, 1)

• (-∞, -1)

• (-1, ∞)

• (1, ∞)

What is the final domain of f(g(x))?

• x ≤ -1

• x = -1

• x > -1

• x < -1