Understanding Inverse Functions and Domains

The video tutorial explains how to find the inverse of a given function g(x) = tan(x - 3π/2) + 6 and determine its domain. It involves solving for g inverse by swapping variables and using inverse tangent properties. The tutorial also covers the domain and range of tangent and inverse tangent functions, emphasizing the need to restrict the domain for invertibility. Finally, it concludes with the domain and range of g inverse, highlighting that the domain is all real numbers, while the range is from π to 2π.

What is the initial function given in the problem?

• g(x) = cot(x) - 3π/2 + 6

• g(x) = cos(x) - 3π/2 + 6

• g(x) = tan(x) - 3π/2 + 6

• g(x) = sin(x) - 3π/2 + 6

What is the first step in finding the inverse of a function?

• Add a constant to both sides

• Swap the variables and solve for the new dependent variable

• Take the derivative of the function

• Multiply both sides by a constant

What operation is performed after subtracting 6 from both sides in the process of finding the inverse?

• Taking the square root

• Taking the inverse tangent

• Multiplying by 2

• Adding 3π/2

What is the domain of the tangent function?

• All real numbers except multiples of π

• All real numbers except multiples of 2π

• All real numbers

• All real numbers except multiples of π/2

Why is it necessary to restrict the domain of the tangent function?

• To make it continuous

• To make it differentiable

• To make it invertible

• To make it periodic

What is the range of the inverse tangent function?

• Between 0 and π

• Between 0 and 2π

• Between -π/2 and π/2

• Between -π and π

What is the domain of the inverse tangent function?

• Only integers

• All real numbers

• Only positive numbers

• Only negative numbers

What is the domain of the inverse function g inverse?

• Negative infinity to positive infinity

• Zero to infinity

• Zero to one

• Negative infinity to zero

What is the range of the inverse function g inverse?

• 0 to 2π

• π/2 to 3π/2

• 0 to π

• π to 2π

What is the significance of the range of the inverse function g inverse?

• It includes all real numbers

• It is an open interval from π to 2π

• It includes only positive integers

• It is a closed interval from π to 2π