Understanding Domain, Range, and Inverse Functions

The video tutorial explains the function f(x) = 4 - x^2 with a domain restriction of x ≥ 0. It covers how to express the domain and range using interval notation, find the range of the function, and verify these graphically using Desmos. The tutorial also explains how to find the inverse function by interchanging x and y, solving for y, and ensuring the inverse's domain and range are correct. Finally, it checks the symmetry of the function and its inverse graphically.

What is the domain of the function f(x) = 4 - x^2 when x is restricted to be greater than or equal to 0?

• (-∞, 0]

• (-∞, ∞)

• [0, 4]

• [0, ∞)

How is the range of the function f(x) = 4 - x^2 expressed in interval notation?

• (-∞, 0]

• [4, ∞)

• [0, 4]

• (-∞, 4]

What is the maximum value of the function f(x) = 4 - x^2?

• 4

• 2

• Infinity

• 0

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

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

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

• They are unrelated.

• They are the same.

What is the first step in finding the inverse of the function f(x) = 4 - x^2?

• Solve for x.

• Graph the function.

• Replace f(x) with y.

• Find the derivative.

Why is the negative square root dropped when finding the inverse of f(x) = 4 - x^2?

• Because the domain is restricted.

• Because the range is always non-negative.

• Because it simplifies the equation.

• Because it is not needed for symmetry.

What is the inverse function of f(x) = 4 - x^2?

• f⁻¹(x) = √(4 - x)

• f⁻¹(x) = 4 - √x

• f⁻¹(x) = x² - 4

• f⁻¹(x) = -√(4 - x)

What is the purpose of graphing the function and its inverse on Desmos?

• To calculate the derivative.

• To determine the domain.

• To verify symmetry across y = x.

• To find the maximum value.

Which line should the function and its inverse be symmetrical across?

• x = 0

• y = x

• y = 0

• x = y

What color is used to graph the inverse function in Desmos?

• Green

• Black

• Blue

• Red