Problem 55 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x+2}{x} \) for \( x > 0 \).

Step 1: Understand the Inverse Function

An inverse function, denoted as \( f^{-1} \), is a function that reverses the effect of the original function \( f \). If \( f(x)=y \), then \( f^{-1}(y)=x \). Therefore, finding the inverse involves swapping the dependent and independent variables and solving for the independent variable.

Step 2: Swap and Set Equation

Start with the equation for \( f(x) \): \( y = \frac{2}{x - 1} \). Next, swap \( x \) and \( y \), giving us \( x = \frac{2}{y - 1} \). This prepares us to solve for \( y \).

Step 3: Solve for the Inverse Function

To find \( y \), first multiply both sides by \( y - 1 \): \( x(y - 1) = 2 \). Simplify the expression: \( xy - x = 2 \). Add \( x \) to both sides to get: \( xy = x + 2 \). Then divide each term by \( x \): \( y = \frac{x + 2}{x} \). Thus, the inverse function is \( f^{-1}(x) = \frac{x + 2}{x} \) for \( x > 0 \).

Step 4: Graph the Original Function

Graph \( f(x) = \frac{2}{x-1} \) by recognizing that it is a rational function with a vertical asymptote at \( x = 1 \) and a horizontal asymptote at \( y = 0 \). The function is defined for \( x > 1 \) and approaches the asymptotes without crossing them.

Step 5: Graph the Inverse Function

Graph the inverse function \( f^{-1}(x) = \frac{x+2}{x} \). Recognize that it also has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 1 \). This function is defined for \( x > 0 \). The graphs of \( f \) and \( f^{-1} \) should reflect across the line \( y = x \).

Step 6: Verify Graphically

Make sure both functions \( f(x) \) and \( f^{-1}(x) \) are symmetric with respect to the line \( y = x \) in the graph. This symmetry is a crucial feature of functions and their inverses, verifying correctness.

Key Concepts

Graphing Functions

Graphing functions is an essential skill in mathematics, as it helps us visualize how a function behaves. When graphing any function, always start by identifying key features like intercepts, asymptotes, and the domain of the function.

For the function given, \[ f(x) = \frac{2}{x - 1} \],we first notice that the function is only defined for \( x > 1 \) because \( x = 1 \) will make the denominator zero, leading to undefined behavior.

• **Domain:** \( x > 1 \)
• **Vertical Asymptote:** \( x = 1 \)
• **Horizontal Asymptote:** Approaches \( y = 0 \) as \( x \) goes to infinity.

Visualizing these elements on a set of axes helps to understand how \( f(x) \) behaves and understand its limitations. The graph approaches the vertical asymptote but never crosses it, demonstrating how graphs can approach but not touch certain lines.

Rational Functions

Rational functions are composed of a ratio of two polynomials. They often have intriguing behaviors because of the relationships between their numerator and denominator.

In our example, \[ f(x) = \frac{2}{x - 1} \],we see that it is a simple rational function with the numerator as a constant \( 2 \) and the denominator \( x-1 \). This function showcases the essential property of rational functions having asymptotes where the denominator equals zero, which heavily influences the function's behavior

The inverse function \[ f^{-1}(x) = \frac{x + 2}{x} \],is another rational function with different components creating new asymptotes.

• This function has a vertical asymptote at \( x = 0 \) since the denominator would make the function undefined.
• It also has a horizontal asymptote at \( y = 1 \), showcasing a shift compared to the original function.

Understanding the structure of rational functions is crucial for anticipating their behavior both algebraically and graphically.

Asymptotes

Asymptotes are lines a graph approaches but never actually reaches. They provide a framework for understanding the boundaries and behavior of a function as the input values become very large or very small.

In our exercise, both functions \( f(x) = \frac{2}{x-1} \) and \( f^{-1}(x) = \frac{x+2}{x} \) have asymptotes that help define their behavior.

• For \( f(x) \), there is a vertical asymptote at \( x = 1 \), since the denominator would become zero here.
• A horizontal asymptote is observed at \( y = 0 \), meaning as \( x \) goes to infinity, the function value approaches zero.
• For the inverse function \( f^{-1}(x) \), the vertical asymptote is at \( x = 0 \) and a horizontal asymptote at \( y = 1 \).

These asymptotes guide us in properly sketching the functions and understanding how their values behave as \( x \) increases or decreases significantly.