Problem 56 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=\frac{-1}{x-2} \quad \text { for } x>2 $$

Step-by-Step Solution

Answer

The inverse of \( f(x) = \frac{-1}{x-2} \) is \( f^{-1}(x) = \frac{-1}{x} + 2 \).

Step 1: Solve for the Inverse Function

To find the inverse of the function \( f(x) = \frac{-1}{x-2} \), we start by replacing \( f(x) \) with \( y \). So the equation becomes \( y = \frac{-1}{x-2} \). Now, we solve for \( x \) in terms of \( y \). Multiply both sides by \( x-2 \) to get \( y(x-2) = -1 \). Next, solve for \( x \) to find: \( x = \frac{-1}{y} + 2 \). Thus, the inverse function is \( f^{-1}(x) = \frac{-1}{x} + 2 \).

Step 2: Analyze the Graph of the Original Function

The original function \( f(x) = \frac{-1}{x-2} \) is defined for \( x > 2 \). It includes a vertical asymptote at \( x = 2 \) since the function is undefined at this value. There is a horizontal asymptote at \( y = 0 \), as the function approaches 0 when \( x \) moves towards infinity. The function decreases as \( x \) increases beyond 2.

Step 3: Graph the Inverse Function

The inverse function \( f^{-1}(x) = \frac{-1}{x} + 2 \) has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 2 \). This function is defined for all \( x eq 0 \), but since \( f(x) \) is defined only for \( x > 2 \), \( f^{-1}(x) \) will be meaningful only for \( y \) values obtained from \( x > 2 \), resulting in \( f^{-1} \) being defined for \( x > 0 \).

Step 4: Plot Both Functions on the Same Set of Axes

To present the functions on the same graph, plot \( f(x) = \frac{-1}{x-2} \) for \( x > 2 \) and \( f^{-1}(x) = \frac{-1}{x} + 2 \) for \( x > 0 \). Both graphs should illustrate their respective asymptotes, and where \( f \) is plotted with points higher on the y-axis as \( x \) increases, \( f^{-1} \) should have a range starting just above \( x = 0 \). The graph of \( f \) can be represented as moving further away from \( x = 2 \), while \( f^{-1} \) approaches \( y = 2 \).

Key Concepts

Graphing Functions

When it comes to graphing functions, especially the original function and its inverse, visualizing can bring a clearer understanding of their behavior. Let's explore how to sketch these functions on the same set of axes. To begin with, our original function is \( f(x) = \frac{-1}{x-2} \), defined for \( x > 2 \). This function shifts slightly off the axes, and drawing it requires careful attention to its specific range and domain constraints.

• Set up your axes by marking the point where the function is undefined, \( x = 2 \), as a vertical asymptote.
• As \( x \) increases past 2, note how the function decreases, giving it a trailing curve that approaches \( y = 0 \), the horizontal asymptote.

Next, focusing on the inverse \( f^{-1}(x) = \frac{-1}{x} + 2 \), we account for its own shift and domain: for any real number \( x eq 0 \), but only meaningful here on \( x > 0 \).

• Add a vertical asymptote at \( x = 0 \).
• The horizontal asymptote shifts to \( y = 2 \).
• Sketch a curve beginning above \( x = 0 \), where it wraps gently around \( y = 2 \) as \( x \) grows.

Combining both graphs provides a comprehensive view, allowing you to pinpoint their transformation relationship, observing them as reflections over the line \( y = x \). Each curve subtly mirrors movement about this line, demonstrating the transformative effects of function inversion.

Asymptotes

In mathematics, asymptotes are crucial in understanding the behavior of curves, particularly in rational functions like our example. Let's delve into the types of asymptotes unveiled in our original and inverse functions.

• Vertical Asymptotes: In the case of \( f(x) = \frac{-1}{x-2} \), the vertical asymptote is at \( x = 2 \). This asymptote represents where the function heads towards infinity, never quite touching or crossing this line but moving infinitely closer as \( x \) approaches 2. For the inverse, \( f^{-1}(x) = \frac{-1}{x} + 2 \), a vertical asymptote appears at \( x = 0 \). This marks the division point where the function cannot take any real value since the denominator becomes zero.
• Horizontal Asymptotes: Often, horizontal asymptotes indicate the behavior of a function as \( x \) approaches infinity. In our original function, \( f(x) = \frac{-1}{x-2} \) approaches \( y = 0 \), representing the value the function tends to, where output gradually diminishes closer towards zero. Conversely, the inverse \( f^{-1}(x) = \frac{-1}{x} + 2 \) reveals a horizontal asymptote at \( y = 2 \), signaling a higher stable approach point.

Understanding these asymptotic lines helps us comprehend the ultimate behavior and limits of functions beyond the visible graph, giving insight into what happens as the functions stretch towards infinity or collapse into their limits.

Function Analysis

Function analysis encompasses the study of different aspects of functions to fully understand their behavior and properties. Using the functions \( f(x) = \frac{-1}{x-2} \) and its inverse \( f^{-1}(x) = \frac{-1}{x} + 2 \), we can dissect various features essential to grasping their full scope.

• Domain and Range: For any function, defining domain and range is critical. The domain of \( f(x) \) starts just beyond \( x = 2 \) (\( x > 2 \)), due to its vertical asymptote. Its range aligns similarly to \( y \) values approaching zero yet remaining negative. On the other hand, when examining \( f^{-1}(x) \), its domain is any \( x > 0 \) since only these values correspond to where the original function's outputs could occur. The result is a range of any \( y eq 2 \) values.
• Behaviors and Trends: Recognizing how these functions grow or shrink is insightful. As \( f(x) \) progresses with increasing \( x \), it demonstrates decreasing values, nearing the x-axis asymptote but never reaching it. Meanwhile, \( f^{-1}(x) \) reveals increasing values as \( y \) moves towards 2 unintersected.
• Reflection and Symmetry: The concept of inverse functions naturally introduces a reflective trait over the line \( y = x \). This symmetry ensures that if the original function gives specific output, its inverse will navigate back through this point, highlighting the reciprocal, mirrored structure.

Delving into these elements gives a robust understanding of how mathematical entities transition and reflect, ultimately leading to enhanced problem-solving skills and graphical interpretation.