Cubic Functions and Their Inverses

This video tutorial covers the concept of inverse functions, focusing on quadratic and cubic functions. It explains many-to-one functions, the vertical line test, and how to restrict the domain to ensure the inverse is a function. The tutorial includes graphing exercises for both quadratic and cubic functions, demonstrating how to find and graph their inverses. The video concludes with instructions for assignments, emphasizing the importance of understanding reflections over the line y = x.

What is the purpose of the vertical line test?

• To determine if a graph is a function

• To find the inverse of a function

• To identify the domain of a function

• To calculate the slope of a line

Why do we restrict the domain of a quadratic function?

• To increase the range of the function

• To simplify the graphing process

• To ensure its inverse is a function

• To make it pass the horizontal line test

How do you find the inverse of a quadratic function?

• Switch the x and y values and solve for y

• Multiply the function by -1

• Add a constant to the function

• Divide the function by 2

What is the result of reflecting a function over the line y = x?

• A horizontal line

• A vertical line

• The original function

• The inverse function

In the context of inverse functions, what does swapping x and y achieve?

• It reflects the graph over the x-axis

• It reflects the graph over the y-axis

• It reflects the graph over the line y = x

• It rotates the graph 90 degrees

What shape does the graph of a cubic function typically have?

• A circle

• An S-curve

• A straight line

• A parabola

How does a vertical compression affect a cubic function's graph?

• It compresses the graph horizontally

• It stretches the graph vertically

• It compresses the graph vertically

• It stretches the graph horizontally

What is the inverse of the function y = x^3 - 9?

• y = (x + 9)^3

• y = cube root(x + 9)

• y = x^3 + 9

• y = cube root(x - 9)

What is the effect of a negative coefficient in a cubic function?

• It shifts the graph upwards

• It stretches the graph horizontally

• It compresses the graph vertically

• It reflects the graph over the x-axis

What is the general form of a cubic function?

• y = a/x

• y = ax + b

• y = ax^3 + bx^2 + cx + d

• y = ax^2 + bx + c