Exploring Inverse Functions with Restricted Domains

The video tutorial explains inverse relations and how to ensure their graphs are functions. It begins with a review of inverse relations and the importance of one-to-one functions. The teacher demonstrates graphing a parabola and checking if it is one-to-one. To make the inverse a function, the domain is restricted. The process of finding and graphing the inverse function is shown, highlighting the switch of domain and range between the original and inverse functions.

What did we learn about inverse relations in section 2.7?

• They always result in a function

• They do not always result in a function if the original is not one-to-one

• They are easier to graph than original functions

• They do not exist

What is the vertex of the parabola for the function x^2 + 3?

• (3, 3)

• (0, 3)

• (3, 0)

• (0, 0)

How do we test if a function is one-to-one?

• Using a vertical line test

• By graphing it

• By finding its inverse

• Using a horizontal line test

Why is the function x^2 + 3 not one-to-one?

• It passes the horizontal line test

• It is not a polynomial

• It fails the vertical line test

• It fails the horizontal line test

What is the common method to make a function one-to-one?

• Differentiate the function

• Restrict its range

• Square the function

• Restrict its domain

Which side of the parabola is commonly used when restricting the domain to make the function one-to-one?

• Both sides

• Neither side

• Positive side

• Negative side

What is the new domain of the function when only the positive side of the parabola is used?

• Negative infinity to zero

• Zero to positive infinity

• Negative infinity to positive infinity

• Zero to three

What is the first step in finding the inverse of the function f(x) = x^2 + 3?

• Graph the function

• Add 3 to both sides

• Square root both sides

• Switch x and y and solve for y

Why do we not need the negative square root when finding the inverse of the restricted function?

• The function is linear

• The inverse does not exist

• The function is not differentiable

• The domain only includes positive x values

What happens to the domain and range when finding the inverse of a function?

• They remain unchanged

• They become undefined

• They both become infinite

• They switch places