Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Show that the graph of \(f^{-1}\) is the reflection of the graph of \(f\) through the line \(y=x\) by verifying the following conditions: (1) If \(P(a, b)\) is on the graph of \(f\), then \(Q(b, a)\) is on the graph of \(f^{-1}\). (2) The midpoint of line segment \(P Q\) is on the line \(y=x\). (3) The line \(P Q\) is perpendicular to the line \(y=x\).

Step-by-Step Solution

Answer

The graph of \(f^{-1}\) is the reflection of \(f\) across \(y=x\) as shown by point reflection, perpendicularity, and midpoint on \(y=x\).

Step 1: Understanding Point Reflection

The function \( f \) and its inverse \( f^{-1} \) have a reflective relationship over the line \( y = x \). If \( P(a, b) \) is a point on the graph of \( f \), then for the inverse \( f^{-1} \), point \( Q(b, a) \) should exist. This is because inversion switches the coordinates \((a, b)\) to \((b, a)\). Thus, if \( f(a) = b \), then \( f^{-1}(b) = a \). This confirms condition (1).

Step 2: Finding the Midpoint of Segment PQ

Point \( P(a, b) \) is on the graph of \( f \), and point \( Q(b, a) \) is on the graph of \( f^{-1} \). The midpoint \( M \) of segment \( PQ \) can be found using the midpoint formula: \[ M = \left( \frac{a+b}{2}, \frac{b+a}{2} \right) \]. Since \( \frac{a+b}{2} = \frac{b+a}{2} \), the midpoint lies on the line \( y = x \), thus proving condition (2).

Step 3: Checking Perpendicularity of Segment PQ to Line y=x

The slope of the line \( y=x \) is 1. The slope of line segment \( PQ \) connecting \( P(a, b) \) to \( Q(b, a) \) is calculated as: \[ m_{PQ} = \frac{a-b}{b-a} = -1 \]. Two lines are perpendicular if the product of their slopes is \(-1\). Thus, \( 1 \times (-1) = -1 \), which confirms that line \( PQ \) is perpendicular to the line \( y=x \), satisfying condition (3).

Key Concepts

Graph Reflection

Understanding graph reflections can greatly aid in grasping the concept of inverse functions. Graph reflection, in this context, refers to flipping a graph over a specific line, which, for inverse functions, is the line \( y = x \). If you have a function \( f \) and its inverse \( f^{-1} \), any point \( P(a, b) \) that lies on \( f \) will have a corresponding point \( Q(b, a) \) on \( f^{-1} \). This reflection swaps the x and y coordinates, flipping the graph over the line \( y = x \). Such a transformation leads to a mirrored image of the graph across this line.
By showing that point \( Q(b, a) \) is part of the graph of \( f^{-1} \), you confirm that the graph \( f^{-1} \) is indeed the reflection of \( f \) through \( y=x \). Understanding how to identify these corresponding points can simplify the visualization of function inverses.

Midpoint Theorem

The Midpoint Theorem is a useful tool in confirming properties of inverse functions. The theorem states that the midpoint \( M \) of a segment with endpoints \( P(a, b) \) and \( Q(b, a) \) can be calculated using the formula:

• \( M = \left( \frac{a+b}{2}, \frac{b+a}{2} \right) \)

Using this formula, when you find that \( M \) results in identical x and y values, it's located on the line \( y = x \). This shows reflection symmetry since any such midpoint must lie on \( y = x \).
Thus, the Midpoint Theorem assists in verifying that the graph of \( f^{-1} \) is not only a reflection over \( y = x \) based on points \( (a, b) \) and \( (b, a) \), but also that each segment formed is perfectly bisected by this line through this midpoint position.

Perpendicular Lines

Perpendicular lines intersect at a right angle, which in terms of slopes, means the product of their slopes is \(-1\). For the lines to be perpendicular, if one slope is \(1\), the other must be \(-1\). When assessing points \( P(a, b) \) and \( Q(b, a) \), the line segment between these points has slope

• \( m_{PQ} = \frac{a-b}{b-a} = -1 \)

The line \( y = x \) naturally has a slope of \( 1 \). Thus,
\( m_{PQ} \cdot 1 = -1 \), confirming that line \( PQ \) is indeed perpendicular to \( y = x \). This perpendicular relationship is vital as it provides geometric evidence that lines meet at a right angle, further proving the reflection of the graph.

Coordinate Geometry

Coordinate geometry, or analytical geometry, allows us to describe geometric figures using algebra through coordinates. When working with graph reflections and inverses:

• Coordinate geometry gives us the tools to switch points from \( (a, b) \) to \( (b, a) \).
• It helps verify midpoints using algebraic formulae.
• It allows calculation of slopes to check perpendicularity.

Understanding these processes through coordinate geometry provides clarity in the relationship between a function and its inverse.
This foundational concept ensures that any point on an inverse function can be finitely derived, ensuring proper understanding of graph transformations and symmetries that occur with reflecting functions over key axes, like \( y = x \).