Function Inverses and Compositions

The video tutorial explains how to determine if two functions are inverses by checking if the composition of the functions equals the identity function in both orders. It provides multiple examples, including quadratic, linear, rational, and cubic functions, to illustrate the process of verifying inverse relationships.

What is the main criterion to determine if two functions are inverses of each other?

• Their graphs intersect at the origin.

• They are both linear functions.

• Their compositions equal the identity function.

• They have the same domain.

In the example where f(x) = x^2 + 5 and g(x) = √(x - 5), what is the result of f(g(x))?

• x + 5

• x - 5

• x

• x^2

Why are the functions f(x) = 3x + 8 and g(x) = 8x^2 - 3 not inverses?

• Their graphs do not intersect.

• They are not both quadratic.

• They have different domains.

• Their compositions do not simplify to x.

What is the result of the composition f(g(x)) for f(x) = 7x + 5 and g(x) = (x - 5)/7?

• x

• x + 5

• 7x

• x - 5

How can you verify that f(x) = 7x + 5 and g(x) = (x - 5)/7 are inverses?

• By checking if f(g(x)) = x and g(f(x)) = x.

• By ensuring their graphs are parallel.

• By checking if they are both linear.

• By confirming they have the same range.

In the practice problem, what is the result of g(f(x)) for f(x) = 5/x + 3 and g(x) = 5/(x - 3)?

• x

• x - 3

• x + 3

• 5x

What operation is performed to simplify the complex fraction in the practice problem?

• Addition

• Subtraction

• Multiplication

• Division

In the final example, what is the result of f(g(x)) for f(x) = x^3 + 5 and g(x) = ∛(x - 5)?

• x^3

• x

• x - 5

• x + 5

What is the inverse of the function f(x) = x^3 + 5?

• x^3 - 5

• ∛(x + 5)

• ∛(x - 5)

• x^3 + 5

What is the result of g(f(x)) for f(x) = x^3 + 5 and g(x) = ∛(x - 5)?

• x^3

• x

• x + 5

• x - 5