Inverse Functions and Their Properties

The video tutorial explains how to find inverse functions for given one-to-one functions. It covers the concept of inverse functions, demonstrating how they undo each other. The tutorial provides step-by-step instructions to find inverse functions, including interchanging variables and solving equations. Graphical methods are used to verify the correctness of inverse functions by checking symmetry across the line y = x. Two examples are provided: one involving a rational function and another with a cube root function.

What is a key property of one-to-one functions that allows them to have inverses?

• They are always linear.

• They have a unique output for each input.

• They are always quadratic.

• They have multiple outputs for each input.

When finding the inverse of a function, what is the first step?

• Replace y with x.

• Replace f(x) with y.

• Multiply by the reciprocal.

• Add a constant to both sides.

In the example given, what operation is performed after interchanging x and y to solve for y?

• Divide both sides by x.

• Add 5 to both sides.

• Subtract 5 from both sides.

• Multiply both sides by 4.

How can you verify graphically that two functions are inverses of each other?

• They are symmetrical across the line y = x.

• They are parallel to each other.

• They intersect at the origin.

• They have the same slope.

What is the first step in finding the inverse of the cube root function given in the second example?

• Square both sides.

• Replace f(x) with y.

• Add 3 to both sides.

• Multiply by the cube root.

What operation is used to eliminate the cube root when solving for y in the second example?

• Divide by 3.

• Cube both sides.

• Take the square root of both sides.

• Square both sides.

After cubing both sides in the second example, what is the next step to solve for y?

• Divide by 3.

• Multiply by 3.

• Subtract 3 from both sides.

• Add 3 to both sides.

What is the inverse function of f(x) = cube root of (x + 3)?

• f inverse of x = cube root of (x + 3)

• f inverse of x = cube root of (x - 3)

• f inverse of x = x^3 - 3

• f inverse of x = x^3 + 3

What is the significance of the line y = x in verifying inverse functions?

• It is the axis of symmetry for the functions.

• It is the line where the functions intersect.

• It is the line where the functions are perpendicular.

• It is the line where the functions are parallel.

What does the symmetry across the line y = x indicate about two functions?

• They have the same domain.

• They are inverses of each other.

• They are identical.

• They have the same range.