Determining Functions and Decomposition

The video tutorial explains how to solve a function problem involving composite functions. It begins with an introduction to the problem, where FX and GX are given, and the composite function G(FX) is provided. The tutorial then delves into understanding composite functions and the process of decomposition. The instructor provides a step-by-step solution to find GX by decomposing the given composite function. The video concludes with the final steps and a summary of the solution, ensuring clarity in understanding the process.

What is the main objective of the problem discussed in the video?

• To calculate the derivative of g(x)

• To find the value of x

• To determine the function g(x)

• To solve for f(x)

What is the given expression for f(x)?

• f(x) = x^2 - 1

• f(x) = 3x - 2

• f(x) = 2x + 3

• f(x) = 5x + 4

What is the expression for g(f(x)) provided in the problem?

• g(f(x)) = 2x - 5

• g(f(x)) = 6x - 1

• g(f(x)) = 5x + 2

• g(f(x)) = 4x + 3

What is the first step in decomposing the composite function?

• Differentiate g(f(x))

• Integrate f(x)

• Substitute f(x) into g(f(x))

• Solve for x

What is the expression for g(x) after substituting f(x) into g(f(x))?

• g(x) = 6x - 1

• g(x) = 2(3x - 2) + 3

• g(x) = 3x - 2

• g(x) = 2x + 1

What operation is performed to equate the expressions for g(f(x))?

• Division

• Multiplication

• Subtraction

• Addition

What is the value added to balance the equation for g(x)?

• 3

• 2

• 1

• 4

What is the final expression for g(x) after replacing variables?

• g(x) = 6x - 1

• g(x) = 2(3x - 2) + 3

• g(x) = 3x - 2

• g(x) = 2x + 3

What is the purpose of replacing variables in the final step?

• To solve for x

• To ensure consistency with f(x)

• To find the derivative

• To simplify the expression

What is the role of the constant added in the equation?

• To find the intercept

• To balance the equation

• To solve for x

• To adjust the slope