Problem 86 - Intermediate Algebra

Question

a. The graph of a one-to-one function lies entirely in quadrant I. In what quadrant does the graph of its inverse lie? b. The graph of a one-to-one function lies entirely in quadrant II. In what quadrant does the graph of its inverse lie? c. The graph of a one-to-one function lies entirely in quadrant III. In what quadrant does the graph of its inverse lie? d. The graph of a one-to-one function lies entirely in quadrant IV. In what quadrant does the graph of its inverse lie?

Step-by-Step Solution

Answer

a. Quadrant I, b. Quadrant IV, c. Quadrant III, d. Quadrant II.

Step 1: Understand the concept of inverse functions and quadrants

For a one-to-one function, the graph of its inverse function is a reflection over the line \( y = x \). This means if the original function is in one quadrant, its inverse will be in the quadrant on the opposite side of \( y = x \). The four quadrants are identified as: Quadrant I (positive x, positive y), Quadrant II (negative x, positive y), Quadrant III (negative x, negative y), and Quadrant IV (positive x, negative y).

Step 2: Find the inverse quadrant for part a

Given that the graph of the function is entirely in Quadrant I, where both x and y are positive, the inverse graph will also need positive x and positive y since reflections yield the same. Thus, the inverse is also in Quadrant I.

Step 3: Find the inverse quadrant for part b

For a function in Quadrant II, where x is negative and y is positive, the inverse graph will have x positive and y negative (since it reflects over \( y = x \)), placing it in Quadrant IV.

Step 4: Find the inverse quadrant for part c

With the function in Quadrant III, where both x and y are negative, the inverse will also have both x and y negative, so it will remain in Quadrant III.

Step 5: Find the inverse quadrant for part d

For a function in Quadrant IV, where x is positive and y is negative, the inverse will have x negative and y positive, thus placing it in Quadrant II.

Key Concepts

One-to-One Functions

A one-to-one function is an important concept in mathematics, especially when exploring inverse functions. Such a function has a special property: each unique input (x-value) maps to a unique output (y-value), which means no two different inputs result in the same output. This characteristic is crucial when determining whether a function has an inverse because only one-to-one functions are guaranteed to have inverses that are also functions. In practical terms, to check if a function is one-to-one, you can use the horizontal line test. If every horizontal line drawn across the graph of the function touches it at most once, the function is one-to-one. Applying this test helps to visually confirm whether an inverse can exist for a given function.

Graph Reflection

Inverse functions have a unique graphical property; they reflect across the line represented by the equation \(y = x\). This line essentially acts as a mirror, flipping the coordinates of the original function's points to create a new shape on the graph. To comprehend reflection, consider a point \((a, b)\) on the original function. Its reflection over \(y = x\) would be \((b, a)\) on the inverse function. This mirroring property is why the graphical appearance of an inverse function looks like a mirror image around the line \(y = x\). This reflection helps in visually identifying the inverse on a graph, and also assists in finding the correct quadrant placement for inverse graphs, as seen in the solution to the exercise questions.

Cartesian Quadrants

The Cartesian coordinate system is divided into four sections called quadrants, each with different sign combinations for the x and y axes.

• Quadrant I: both x and y are positive.
• Quadrant II: x is negative while y is positive.
• Quadrant III: both x and y are negative.
• Quadrant IV: x is positive and y is negative.

Understanding these quadrants is crucial when determining the location of a function or its inverse in graphical exercises. When a function’s graph resides entirely in one quadrant, the inverse's position will depend on its reflection over the line \(y = x\). For instance, if the function is in Quadrant III, the inverse after reflection will still be in Quadrant III because both x and y coordinates remain negative, just as shown in the exercise solution.