Composite Functions and Simplification

This video tutorial explains composite functions using function machines. It starts with defining two functions, f(x) and g(x), and demonstrates how to connect them to form a composite function. The video provides examples of calculating g(f(x)) and f(g(x)), showing step-by-step simplification. It also covers the notation used for composite functions and offers additional examples for practice. The tutorial concludes with a recap and encourages viewers to subscribe for more content.

What is a composite function?

• A function that is composed of two or more functions.

• A function that only has one variable.

• A function that is always linear.

• A function that cannot be simplified.

What does a function machine represent?

• A conceptual model to understand how functions transform inputs to outputs.

• A tool for solving differential equations.

• A physical machine that calculates functions.

• A device used in calculus.

How do you combine two function machines?

• By adding their outputs.

• By using the output of one as the input for the other.

• By multiplying their outputs.

• By subtracting their outputs.

What is the output of the composite function when f(x) = 3x + 2 and g(x) = x^2 - 1?

• 3x^2 + 2x - 1

• (3x + 2)^2 - 1

• 3x^2 - 1

• x^2 + 3x + 1

What is the first step in simplifying a composite function?

• Divide all terms by 3.

• Multiply all terms by 2.

• Replace the variable with brackets.

• Add 5 to all terms.

What is the result of expanding (x + 1)^2?

• x^2 + 2

• x^2 + 1

• x^2 + 1x

• x^2 + 2x + 1

What is the simplified form of g(f(x)) when g(x) = x^2 - 5 and f(x) = x + 1?

• x^2 + 2x - 4

• x^2 - 5

• x^2 + 2x - 5

• x^2 + 2x + 1

What is the purpose of using brackets in composite functions?

• To divide the function.

• To add complexity.

• To replace variables for simplification.

• To indicate multiplication.

What is the result of subtracting 5 from 1?

• -4

• 4

• 0

• -5

What is the result of adding 1 to x^2?

• x^2 + 0

• x^2 + 2

• x^2 + x

• x^2 + 1