Understanding Composite Functions

This video tutorial reviews the concept of composite functions, providing examples and exercises to help understand how functions can be composed. It explains the process of composing functions, using examples to illustrate the steps involved. The tutorial also discusses how to express functions as compositions of other functions and explores the difference between products and compositions of functions. The goal is to build foundational skills necessary for calculus, particularly the chain rule.

What is the primary focus of composite functions in calculus?

• To understand the chain rule

• To simplify algebraic expressions

• To solve linear equations

• To graph quadratic functions

If f(x) = 1 + x and g(x) = cosine(x), what is f(g(x))?

• x + cosine(x)

• 1 + cosine(x)

• cosine(x) + 1

• 1 + x

How can g(x) = cosine(sin(x) + 1) be expressed as a composition of two functions?

• u(x) = sin(x), v(x) = cosine(x) + 1

• f(x) = x + 1, g(x) = cosine(sin(x))

• f(x) = sin(x), g(x) = cosine(x) + 1

• u(x) = sin(x), v(x) = cosine(x + 1)

Which of the following is a valid composition of three functions for g(x) = cosine(sin(x) + 1)?

• u(x) = sin(x), w(x) = x + 1, h(x) = cosine(x)

• f(x) = x + 1, g(x) = sin(x), h(x) = cosine(x)

• u(x) = x + 1, v(x) = sin(x), w(x) = cosine(x)

• f(x) = sin(x), g(x) = x + 1, h(x) = cosine(x)

What is the difference between a composition and a product of functions?

• A composition involves adding functions, while a product involves multiplying them.

• A composition involves subtracting functions, while a product involves dividing them.

• A composition involves applying one function to the result of another, while a product involves multiplying their outputs.

• A composition involves dividing functions, while a product involves adding them.

Which of the following is NOT a composition of functions?

• f(x) * g(x)

• f(g(x))

• h(g(f(x)))

• g(f(x))

If f(x) = cosine(x) and g(x) = sin(x), what is f(g(x))?

• cosine(x) * sin(x)

• sin(x) + cosine(x)

• sin(cosine(x))

• cosine(sin(x))

Which function can be expressed as a product rather than a composition?

• f(x) = sin(x) + cosine(x)

• f(x) = cosine(sin(x))

• f(x) = sin(x) * cosine(x)

• f(x) = sin(cosine(x))

What is the result of composing u(x) = sin(x) and v(x) = cosine(x) + 1?

• sin(x) + 1

• cosine(x) + sin(x)

• sin(cosine(x) + 1)

• cosine(sin(x)) + 1

Which of the following expressions represents a composition of functions?

• f(g(x))

• f(x) - g(x)

• f(x) + g(x)

• f(x) * g(x)