The graph of a one-to-one function is shown to the right. Draw the graph of the inverse function f⁻¹. Choose the correct graph of the inverse function f⁻¹ below.
Understand the Problem
The question is asking to draw the graph of the inverse function based on the given graph of a one-to-one function. It describes the behavior and the specific points of the original function, and asks to identify the correct graph of its inverse from the provided options.
Answer
The inverse function graph will reflect the original function across the line $y = x$.
Answer for screen readers
The correct graph of the inverse function, derived from the graph of the original one-to-one function, will be the one that correctly reflects the switched coordinates of the significant points identified.
Steps to Solve
- Identify Points on the Original Graph
Start by identifying significant points (coordinates) on the graph of the original one-to-one function. For example, if the points are $(a, b)$, $(c, d)$, etc., write them down.
- Switch Coordinates for Inverse Function
To find the inverse function, switch the x and y coordinates of each of the significant points identified from the original function. For example, the point $(a, b)$ will become $(b, a)$.
- Plot the New Points
Now plot the new points on the same graph, reflecting these switched coordinates. Ensure that each point is accurately represented according to their new coordinates.
- Draw the Inverse Function Line
Connect the new points smoothly. The shape of the graph for the inverse function should reflect the behavior of the original function, including any features such as increasing or decreasing sections.
- Identify the Correct Graph
Finally, compare your graphed points of the inverse function with the provided options to select the one that matches your newly created inverse graph.
The correct graph of the inverse function, derived from the graph of the original one-to-one function, will be the one that correctly reflects the switched coordinates of the significant points identified.
More Information
Inverses of functions can often be understood graphically. The graph of an inverse function is a reflection of the original function across the line $y = x$. This symmetry illustrates the relationship between a function and its inverse.
Tips
- Forgetting to switch the coordinates. Always ensure to swap x and y values properly.
- Not reflecting the graph properly across the line $y = x$. Remember to look for symmetry in the graph when plotting.
- Misinterpreting the behavior of the original function, which may lead to incorrect graph features for the inverse.
