Question

A function \(f\) has an inverse function \(f^{-1}\). If the graph of \(f\) lies in quadrant I, in which quadrant does the graph of \(f^{-1}\) lie?

Short answer

The graph of \( f^{-1} \) lies in quadrant I.

Step-by-step solution

Understanding the given function and its inverse

A function, denoted as \( f \), maps inputs to outputs in relation to a specific rule. The inverse function, denoted as \( f^{-1} \), reverses this mapping, taking outputs back to corresponding inputs.

Analyzing the graph of the function

The graph of function \( f \) lies in the first quadrant. This means that for all points \( (x, y) \) on the graph, both \( x \) and \( y \) are positive.

Reflecting the graph across the line \( y = x \)

The graph of the inverse function \( f^{-1} \) is a reflection of the graph of \( f \) across the line \( y = x \). Reflecting a point \( (x, y) \) across this line results in the point \( (y, x) \).

Determining the quadrant of the inverse function

Since the coordinates are positive in quadrant I, reflecting these points results in coordinates that remain positive. Hence, the graph of the inverse function \( f^{-1} \) will also lie in quadrant I.

Key concepts

Inverse Function Graph

A function, symbolized as \(f\), follows a rule to map each input to an output. The inverse function, represented as \(f^{-1}\), does the reverse. It takes the output of \(f\) as its input and maps it back to the original input. The graph of an inverse function is very important. If you plot \(f\) on a coordinate plane, the graph of \(f^{-1}\) will show the reverse relationships.
To visualize the inverse function graph, imagine flipping the original function graph across a special line, \(y = x\). This means if you have a point \((a, b)\) on \(f\), you'll find a corresponding point \((b, a)\) on \(f^{-1}\). Understanding this flip or reflection can help you see the nature of inverse functions better.

Quadrants in Coordinate Plane

The coordinate plane is divided into four sections, known as quadrants. They help in identifying the signs of the \(x\) and \(y\) coordinates of points. Here's a quick guide:

• Quadrant I: Both \(x\) and \(y\) are positive.
• Quadrant II: \(x\) is negative, \(y\) is positive.
• Quadrant III: Both \(x\) and \(y\) are negative.
• Quadrant IV: \(x\) is positive, \(y\) is negative.

When dealing with inverse functions, knowing the quadrants can help predict where the inverse will lie. For instance, if \(f\) lies in Quadrant I, then \(f^{-1}\), being a reflection across \(y = x\), will also stay in Quadrant I. This is because positive values remain positive upon reflection.

Reflection Across y=x

Reflection across the line \(y = x\) is a fundamental concept in understanding inverse functions. Imagine this line slicing through the coordinate plane. Any point \((x, y)\) reflected over this line changes to \((y, x)\).
This flip or reflection implies switching the roles of \(x\) and \(y\), which is precisely what an inverse function does. To solve the exercise problem, note that since the graph of \(f\) is in Quadrant I, reflecting it across \(y = x\) means the graph of \(f^{-1}\) will also be in Quadrant I. Both \(x\) and \(y\) values remain positive, which is characteristic of Quadrant I. This visual and conceptual reflection helps understand where graphs of inverse functions will be in relation to their originals.