Exploring Composition of Functions

This video tutorial by Mr. Banker explores the concept of function composition, a mathematical operation that combines two functions. The tutorial provides examples of composing functions, such as f(x) = e^x and g(x) = √x, and discusses the non-commutative nature of composition. It also covers finding the domain of composed functions and demonstrates how to decompose a complex function into simpler components. The video aims to enhance understanding of function operations and their properties.

What is the definition of function composition?

• Dividing one function by another

• Adding two functions together

• Multiplying two functions

• Plugging one function into another

If f(x) = e^x and g(x) = sqrt(x), what is f(g(x))?

• e^(sqrt(x))

• e^x

• sqrt(e^x)

• x^2

What does it mean if function composition is not commutative?

• f(g(x)) = g(f(x))

• f(g(x)) ≠ g(f(x))

• f(x) + g(x) = g(x) + f(x)

• f(x) * g(x) = g(x) * f(x)

What is the domain of the function g(x) = sqrt(x)?

• All real numbers

• x ≥ 0

• x > 0

• x ≤ 0

For f(x) = x^2 - 1 and g(x) = sqrt(x), what is the domain of g(f(x))?

• All real numbers

• x ≤ -1 or x ≥ 1

• x ≥ 0

• x ≤ 0

What is the simplified form of f(g(x)) if f(x) = x^2 - 1 and g(x) = sqrt(x)?

• x^2 - 1

• sqrt(x^2 - 1)

• x - 1

• sqrt(x) - 1

How can you identify the inner function in a composed function?

• By checking the function with the smallest degree

• By identifying the function with the highest degree

• By finding the function inside the parentheses

• By looking for the outermost operation

In the function H(x) = (x + 1)^2 - 3(x + 1) + 4, what is the inner function?

• 4

• 3x

• x + 1

• x^2

For H(x) = sqrt(x^3 + 1), what is the inner function?

• x^3 + 1

• sqrt(x)

• x^2 + 1

• x + 1

What is the outer function in H(x) = sqrt(x^3 + 1)?

• x^3 + 1

• sqrt(x)

• x^2 + 1

• x + 1