Function Decomposition and Composition

The video tutorial explains how to decompose a function f(x) into two functions, h(x) and g(x), which is the reverse of function composition. It provides both basic and creative examples, demonstrating that there are infinite ways to achieve this decomposition. The tutorial encourages creativity and experimentation with different functions to understand the concept better.

What is the reverse operation of composition in functions?

• Multiplication

• Integration

• Differentiation

• Decomposition

When decomposing a function, what are we trying to identify?

• Two functions that compose to form the original function

• The inverse of the function

• The derivative of the function

• The integral of the function

Which of the following is a basic example of decomposing a function?

• h(x) = x^3 and g(x) = x - 2

• h(x) = 1/x and g(x) = x^2

• h(x) = sqrt(x) and g(x) = 3x - 1

• h(x) = x^2 and g(x) = x + 1

In a basic decomposition, if h(x) = x, what should g(x) be to get f(x)?

• g(x) = f(x)

• g(x) = 1/x

• g(x) = x^2

• g(x) = x + 1

What is a creative way to decompose f(x) = 1/(2x - sqrt(x))?

• h(x) = 1/x and g(x) = 2x - sqrt(x)

• h(x) = 2x - sqrt(x) and g(x) = 1/x

• h(x) = x^2 and g(x) = 1/(2x - sqrt(x))

• h(x) = 1/(2x - sqrt(x)) and g(x) = x^2

If h(x) = x + 1, what should g(x) be to achieve f(x) = 1/2x + 3?

• g(x) = x

• g(x) = 1/2x + 2

• g(x) = 2x - 1

• g(x) = x^2

What is the main goal of decomposing a function?

• To find the derivative

• To express it as a composition of two functions

• To find the integral

• To simplify the function

Which of the following is NOT a method of decomposing functions?

• Using logarithmic transformations

• Using creative expressions

• Using trigonometric identities

• Using basic arithmetic operations

What is the result of plugging g(x) into h(x) in a decomposition?

• A simplified version of f(x)

• The inverse of f(x)

• The original function f(x)

• The derivative of f(x)

How many possible decompositions can a function have?

• Infinite

• Two

• One

• None