Graph the inverse of each one-to-one function.

Verified step by step guidance

1. Identify the given function on the graph. The function appears to be a decreasing curve in the first quadrant.
2. To find the inverse, reflect the graph of the function over the line y = x. This means swapping the x and y coordinates of each point on the original graph.
3. Plot the reflected points on the graph. For example, if a point on the original graph is (a, b), plot the point (b, a) on the new graph.
4. Draw a smooth curve through the reflected points to represent the inverse function.
5. Verify that the new graph is a reflection of the original graph over the line y = x, ensuring that each point (a, b) on the original graph corresponds to the point (b, a) on the inverse graph.

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output value is uniquely paired with one input value. This means that no two different inputs produce the same output. This property is essential for a function to have an inverse, as the inverse must also be a function, which requires that each output corresponds to exactly one input.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y and returns x. Graphically, the inverse of a function can be found by reflecting the graph of the original function across the line y = x, which helps visualize the relationship between the function and its inverse.

Graphing Techniques

Graphing techniques involve methods used to accurately represent functions and their inverses on a coordinate plane. This includes plotting points, understanding the shape of the function, and using transformations such as reflections. For one-to-one functions, recognizing the symmetry about the line y = x is crucial for correctly graphing their inverses.