Understanding Composite Functions and Inverses

The video tutorial explains how to determine if two functions are inverses by calculating their composite functions. It starts with defining two functions, f(x) and g(x), and demonstrates how to find f(g(x)) and g(f(x)). The tutorial shows that if both composite functions equal x, the original functions are inverses. A second example with different functions is provided, illustrating the process again and confirming whether the functions are inverses. The video emphasizes the importance of simplifying expressions and verifying both composite functions to determine inverses.

What is the condition for two functions to be inverses of each other?

• Their composite functions equal zero.

• Their composite functions equal x.

• Their composite functions are undefined.

• Their composite functions are equal to each other.

What is the result of f(g(x)) for the functions f(x) = 2x + 3 and g(x) = (x - 3) / 2?

• x

• x + 3

• x - 3

• 2x + 3

In the first example, what is the input for f(x) when finding f(g(x))?

• 2x + 3

• (x - 3) / 2

• x - 3

• x

What is the result of g(f(x)) for the functions f(x) = 2x + 3 and g(x) = (x - 3) / 2?

• 2x + 3

• x

• x - 3

• x + 3

In the first example, what is the input for g(x) when finding g(f(x))?

• (x - 3) / 2

• x - 3

• 2x + 3

• x

What is the result of f(g(x)) for the second example with f(x) = 3x - 9 and g(x) = x / 3 + 9?

• x

• x + 18

• 3x - 9

• x / 3 + 9

In the second example, what is the input for f(x) when finding f(g(x))?

• x

• x / 3 + 9

• 3x - 9

• x + 18

What is the result of g(f(x)) for the second example with f(x) = 3x - 9 and g(x) = x / 3 + 9?

• x

• x / 3 + 9

• 3x - 9

• x + 6

In the second example, what is the input for g(x) when finding g(f(x))?

• x

• 3x - 9

• x / 3 + 9

• x + 6

Why are the functions in the second example not inverses?

• Their composite functions are equal to each other.

• Their composite functions are not equal to x.

• Their composite functions are equal to zero.

• Their composite functions are undefined.