Exploring Inverse Functions and Their Properties

Mr. Tarrou introduces inverse functions, explaining how to determine if two functions are inverses through composition. He demonstrates finding inverses both graphically and algebraically, emphasizing the importance of switching x and y variables and reflecting graphs over the line y=x. The video includes examples and verification of inverse functions using the composition process, ensuring the functions meet the criteria of one-to-one functions.

What is the first step in checking if two functions are inverses?

• Solve for y

• Switch the x and y variables

• Perform the composition f(g(x)) and g(f(x))

• Reflect the graph over the line y=x

What indicates that two functions are inverses of each other?

• f(g(x)) and g(f(x)) are undefined

• f(g(x)) and g(f(x)) are constants

• f(g(x)) = g(f(x)) = y

• f(g(x)) = g(f(x)) = x

How do you graphically find the inverse of a function?

• By switching the x and y intercepts

• By reflecting the graph over the y-axis

• By reflecting the graph over the line y=x

• By rotating the graph 180 degrees

What happens to the coordinates of a point when a graph is reflected over the line y=x?

• The y-coordinate becomes the x-coordinate

• The x-coordinate becomes the y-coordinate

• The coordinates are squared

• The coordinates are unchanged

What is a necessary condition for a function to have an inverse function?

• It must have a y-intercept

• It must pass both the vertical and horizontal line tests

• It must pass the vertical line test only

• It must be a linear function

What does it mean for a function to pass the horizontal line test?

• The function is linear

• Every y-value has a unique x-value

• The function has no inverse

• Every x-value has a unique y-value

How do you algebraically find the inverse of a function?

• By solving for x

• By switching the x and y variables and solving for y

• By taking the reciprocal of the function

• By reflecting the equation over y=x

What is the result of performing the composition f(g(x)) for inverse functions?

• 0

• 1

• y

• x

What does the notation f^-1(x) represent?

• The reciprocal of f(x)

• The inverse of f(x)

• The derivative of f(x)

• The square root of f(x)

Why is it important to check both compositions f(g(x)) and g(f(x)) when verifying inverses?

• To ensure the function is linear

• To confirm the function passes the vertical line test

• To verify the inverse relationship in both directions

• To determine if the function is continuous