Inverse Functions and Their Graphs

This lesson covers inverse functions, focusing on one-to-one correspondence and line tests. It explains how to determine if a function is one-to-one using vertical and horizontal line tests. The lesson also introduces function composition and demonstrates how to evaluate compositions. Finally, it provides a guide on graphing inverse functions by switching ordered pairs and using the line of reflection y = x.

What is a key characteristic of a one-to-one function?

• Each element of the domain corresponds to multiple elements in the range.

• It has repeating values in the domain.

• It fails the vertical line test.

• Each element of the domain corresponds to exactly one element in the range.

Which line test must a function pass to be considered one-to-one?

• Neither vertical nor horizontal line tests

• Vertical line test only

• Horizontal line test only

• Both vertical and horizontal line tests

How do you evaluate the composition h(g(5)) given g(x) = 2x and h(x) = x^2 + 3?

• Add the results of g(5) and h(5)

• Multiply the results of g(5) and h(5)

• Plug 5 into g, then into h

• Plug 5 into h, then into g

What is the result of the triple composition g(h(g(2)))?

• 103

• 38

• 105

• 19

What is the process for graphing an inverse function?

• Translate the graph upwards

• Rotate the graph 90 degrees

• Reflect the graph over the x-axis

• Switch the x and y coordinates of ordered pairs

What is the line of reflection for inverse functions?

• y = 0

• y = x

• y = -x

• x = 0

How does the graph of a parabola change when graphing its inverse?

• It faces to the left

• It remains unchanged

• It faces to the right

• It faces downwards

When graphing the inverse of a line, what happens to the point (2, 0)?

• It maps to (2, -2)

• It maps to (0, 2)

• It maps to (0, 0)

• It maps to (2, 2)

What is a challenge when graphing the inverse of a cubic function?

• Switching x and y coordinates

• Identifying the line of reflection

• Ensuring the graph faces upwards

• Handling multiple bumps in the graph

What is the significance of the line y = x in graphing inverses?

• It is the axis of symmetry

• It is the line of reflection

• It is the horizontal asymptote

• It is the line of best fit