Function Inverses and Domain Restrictions

The video tutorial introduces inverse functions, explaining why x squared lacks an inverse due to failing the vertical line test. It discusses the importance of restricting the domain to pass the horizontal line test, allowing for an inverse function. The tutorial emphasizes choosing the right side of the function for positive values and reviews the process of verifying inverse functions, highlighting its significance in trigonometry.

What is the main reason x squared does not have an inverse function?

• It fails the vertical line test.

• It is not defined for all real numbers.

• It is not a continuous function.

• It has no turning points.

What does the vertical line test determine about a graph?

• Whether it has an inverse.

• Whether it is differentiable.

• Whether it is continuous.

• Whether it is a function or a relation.

How does the horizontal line test relate to the vertical line test?

• It is used to determine continuity.

• It is the inverse of the vertical line test.

• It checks if a function is one-to-one.

• It determines if a function is differentiable.

Why is it important to restrict the domain of a function?

• To make it continuous.

• To ensure it passes the horizontal line test.

• To make it differentiable.

• To simplify its graph.

What feature of a function often causes it to fail the horizontal line test?

• A vertical asymptote.

• A turning point.

• A discontinuity.

• A horizontal asymptote.

What is the inverse function of f(x) = x^2 when the domain is restricted to x ≥ 0?

• f(x) = 1/x

• f(x) = -x

• f(x) = x^2

• f(x) = √x

Why do we prefer to define the square root function as only positive?

• To ensure it is a function.

• To avoid complex numbers.

• To simplify calculations.

• To make it continuous.

What is a practical reason for choosing the positive side of the square root function?

• It is easier to calculate.

• It is more commonly used in real-world applications.

• It avoids negative numbers.

• It is more visually appealing.

What is the result of combining a function with its inverse?

• A new function.

• An undefined result.

• The original input.

• A constant value.

Why is restricting the domain important for trigonometric functions?

• To simplify their graphs.

• To ensure they have inverses.

• To avoid complex numbers.

• To make them continuous.