Understanding Composite Functions and Their Domains

The video tutorial explains how to find the composite function F(G(x)) given F(x) and G(x), and how to determine its domain. It emphasizes the importance of considering the domain of the inner function G(x) and the composite function itself. The tutorial walks through the process of forming the composite function, simplifying it, and solving a quadratic inequality to find the domain. The domain is represented using intervals and inequalities, ensuring the expression under the square root is non-negative.

What must be considered when determining the domain of a composite function?

• Only the domain of the outer function

• Both the domains of the inner and composite functions

• Only the domain of the inner function

• Neither of the domains

What is the domain of the function G(x) = x^2 - 2x?

• Only integers

• Only negative numbers

• Only positive numbers

• All real numbers

How is the composite function F(G(x)) expressed when G(x) = x^2 - 2x and F(x) = sqrt(8 - x)?

• sqrt(8 + x^2 - 2x)

• F(x^2 + 2x)

• sqrt(8 - (x^2 - 2x))

• F(x^2 - 2x)

Why must the expression under the square root in F(G(x)) be non-negative?

• To ensure the function is continuous

• To ensure the function is differentiable

• To ensure the function is integrable

• To ensure the function is defined

What is the first step in solving the quadratic inequality for the domain of the composite function?

• Graphing the function

• Integrating the function

• Factoring a negative out of the expression

• Finding the derivative

What are the critical points found when solving the inequality 8 - x^2 + 2x ≥ 0?

• x = 3 and x = -1

• x = 5 and x = -3

• x = 0 and x = 2

• x = 4 and x = -2

Which interval satisfies the inequality for the domain of the composite function?

• (-∞, -2)

• (-2, 4)

• (4, ∞)

• (-∞, ∞)

How can the domain of the composite function be expressed using inequalities?

• x > -2 and x < 4

• x < -2 or x > 4

• x ≤ -2 or x ≥ 4

• x ≥ -2 and x ≤ 4

What is the graphical representation of the domain of the composite function?

• Open points at -2 and 4

• Closed interval from -∞ to ∞

• Closed points at -2 and 4

• Open interval from -2 to 4

What is the final domain of the composite function F(G(x))?

• (-∞, ∞)

• [-2, 4]

• (-2, 4)

• [0, 8]