Inverse Functions and Their Properties

This tutorial covers the concepts of one-to-one and inverse functions. It begins with an explanation of basic functions, using mapping diagrams to illustrate how domain and range values interact. The tutorial then delves into one-to-one functions, explaining how to identify them using the vertical and horizontal line tests. It introduces inverse functions, detailing how to find and graph them, including reflecting across the line y=x. The tutorial concludes with a discussion on domain-restricted inverse functions, emphasizing the importance of maintaining domain restrictions when dealing with non-one-to-one functions.

What is a function in mathematical terms?

• A relationship where each range value has only one domain value.

• A relationship where each range value has multiple domain values.

• A relationship where each domain value has only one range value.

• A relationship where each domain value has multiple range values.

Which of the following is NOT a function?

• y = 2x

• y = x^2

• y^2 = 2x

• y = 2x^2

What does the vertical line test determine?

• If a function is continuous.

• If a relationship is a function.

• If a function is one-to-one.

• If a function is differentiable.

What is the horizontal line test used for?

• To determine if a function is continuous.

• To determine if a function is differentiable.

• To determine if a function is one-to-one.

• To determine if a relationship is a function.

What is the notation for the inverse of a function f(x)?

• f(x)^1

• 1/f(x)

• f^-1(x)

• f(x)^-1

If f(x) = 5x, what is the inverse function?

• f^-1(x) = 5x

• f^-1(x) = x/5

• f^-1(x) = 1/5x

• f^-1(x) = x^5

How can you graph the inverse of a function?

• Translate it horizontally.

• Rotate it 90 degrees.

• Reflect it across the line y = x.

• Reflect it across the line y = -x.

What is the first step in finding the equation of an inverse function?

• Write the function as y = f(x).

• Solve for y.

• Swap the x and y variables.

• Reflect the graph across the line y = x.

What must be done if the inverse of a function is not one-to-one?

• Reflect the inverse function across the y-axis.

• Change the original function.

• Ignore the inverse function.

• Restrict the domain of the inverse function.

What is the domain of the function f(x) = √x?

• All real numbers.

• x ≥ 0.

• x > 0.

• x ≤ 0.