Inverse Functions and Their Properties

The video tutorial introduces inverse functions, explaining their definition, how to determine if a function has an inverse, and how to find the equation of an inverse function. It emphasizes the importance of one-to-one functions, as only these have inverses. The video provides examples and graphical representations, demonstrating the horizontal line test to identify one-to-one functions. It also explains the formal definition of inverse functions and their properties, including the symmetry of graphs across the line y = x. The tutorial concludes with a summary of key concepts and properties.

What is a one-to-one function?

• A function where each x-value is paired with multiple y-values.

• A function where each y-value is paired with only one x-value.

• A function that fails the vertical line test.

• A function that has no inverse.

Which test helps determine if a function is one-to-one?

• Horizontal line test

• Slope test

• Vertical line test

• Derivative test

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What is the informal relationship between a function and its inverse?

• They add to zero.

• They multiply each other.

• They undo each other.

• They are identical.

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What is the formal definition of an inverse function?

• A function that maps from set B to set A and returns each element to itself.

• A function that maps from set B to set B.

• A function that maps from set A to set B.

• A function that maps from set A to set A.

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What is the first step in determining an inverse function?

• Graph the function.

• Solve for y.

• Interchange x and y variables.

• Check if the function is one-to-one.

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What do you do after interchanging x and y variables to find an inverse?

• Multiply by 2.

• Solve the equation for y.

• Check if the function is one-to-one.

• Graph the function.

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How do you find the inverse of a function given as ordered pairs?

• Multiply each pair by 2.

• Add 1 to each pair.

• Interchange the x and y coordinates.

• Subtract 1 from each pair.

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

• f^-1(x) = (x - 2)^3

• f^-1(x) = x^3 - 2

• f^-1(x) = cube root of (x - 2)

• f^-1(x) = x + 2

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What is the relationship between the domain of a function and its inverse?

• The domain of the function is the opposite of its inverse.

• The domain of the function is unrelated to its inverse.

• The domain of the function is the same as the domain of its inverse.

• The domain of the function is the range of its inverse.

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How are the graphs of a function and its inverse related?

• They are reflections across the line y = x.

• They are parallel to each other.

• They are perpendicular to each other.

• They are identical.

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