Problem 36 - A Graphical Approach to College Algebra

Question

If \(f\) is a function that has an inverse and the graph of \(f\) lies completely within the second quadrant, then the graph of \(f^{-1}\) lies completely within the ____ quadrant.

Step-by-Step Solution

Answer

fourth

Step 1: Understanding Problem Constraints

We are given that the function \( f \) lies completely within the second quadrant. The second quadrant refers to the region of the Cartesian plane where \( x < 0 \) and \( y > 0 \). We aim to find the quadrant where the graph of \( f^{-1} \), the inverse of the function, would lie.

Step 2: Graph of Inverse Function

The inverse function, \( f^{-1} \), swaps the roles of \( x \) and \( y \). This means that the domain and range of \( f \) are exchanged in \( f^{-1} \). For \( f \) in the second quadrant (\( x < 0, y > 0 \)), the inverse function \( f^{-1} \) will have its graph in the region where original \( y \) becomes \( x \) and original \( x \) becomes \( y \).

Step 3: Identifying Quadrant for \( f^{-1} \)

After swapping \( x ext{ and } y \), we will have a graph where \( x > 0 \) and \( y < 0 \). This corresponds to the fourth quadrant, as in the fourth quadrant \( x \) is positive and \( y \) is negative.

Key Concepts

Quadrants

In mathematics, the Cartesian plane is divided into four quadrants. These quadrants help us understand the positioning of points based on the signs of their coordinates, \(x\) and \(y\).

Here's how the quadrants work:

• First Quadrant: Both \(x\) and \(y\) are positive. It's located in the top-right corner.
• Second Quadrant: \(x\) is negative and \(y\) is positive. This is in the top-left corner.
• Third Quadrant: Both \(x\) and \(y\) are negative. You’ll find it in the bottom-left corner.
• Fourth Quadrant: \(x\) is positive, while \(y\) is negative. It's in the bottom-right corner.

For the function \(f\) lying in the second quadrant, the inverse, \(f^{-1}\), will transform and shift to the fourth quadrant.

Function Graphs

Function graphs visually represent the relationship between variables in a function. Understanding these graphs helps us see the behavior and properties of functions.

• For any function \(f\), the graph showcases all input-output pairs \( (x, f(x)) \).
• When considering inverse functions like \(f^{-1}\), the graph swaps its axes. This means every point \( (x, y) \) on \(f\) will appear as \( (y, x) \) on \(f^{-1}\).
• Geometrically, this swapping is a reflection across the line \(y = x\). This line acts like a mirror, flipping the graph over.

For example, if \(f\) is confined to the second quadrant, \(f^{-1}\) will appear entirely in the fourth quadrant due to this symmetric reflection.

Cartesian Plane

The Cartesian plane is a vital tool in mathematics, providing a framework for plotting functions and analyzing their behavior. It consists of two perpendicular axes: the horizontal \(x\)-axis and the vertical \(y\)-axis.

The intersection of these axes creates a grid that helps in positioning points using pairs \( (x, y) \).

• The plane is divided into the four quadrants which are crucial for understanding inverse functions.
• When a function like \(f\) is mapped, its placement in a specific quadrant is defined by its \(x\) and \(y\) signs.
• Inversely, \(f^{-1}\) will follow the opposite rule—flipping the \(x\) and \(y\) coordinates, changing its position accordingly.

Thus, the Cartesian plane not only helps in plotting the original functions but also shows where their inverses will fit after transformation.