Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=3 x$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = x/3\). Both \(f\) and \(f^{-1}\) are linear functions that pass through origin. Their graphs show symmetry to the line \(y=x\).

Step 1: Find the inverse function

To find the inverse function of \(f(x)=3x\), recall that the inverse of a function \(f\) is the function \(f^{-1}\) which reverses the effect of \(f\). This means if \(y = f(x)\) then \(x = f^{-1}(y)\). Thus, to find \(f^{-1}(x)\), set \(f(x)\) equal to \(y\), and then switch \(y\) for \(x\) and vice versa. Doing this, the function becomes \(x = 3y\). Solving this equation for \(y\) gives \(f^{-1}(x) = x/3\).

Step 2: Graph the original and inverse functions

The function \(f(x)=3x\) is a straight line passing through the origin with a slope of 3. This gives a 45 degrees angle with the x-axis. The inverse function \(f^{-1}(x) =x/3\) is also a straight line passing through the origin, but with slope 1/3, which means it's less steep than the original function.

Step 3: Describe the relationship between the graphs

Clearly, \(f\) and \(f^{-1}\) are reflections of each other across the line \(y = x\). This demonstrates the fundamental symmetry property of a function and its inverse. Also, every point on the function \(f(x)\) at coordinate (a,b) corresponds to a point on \(f^{-1}(x)\) at coordinate (b,a).

Key Concepts

Linear functions

Linear functions are integral to understanding inverse functions. They are the simplest form of functions as they create straight lines when graphed. A linear function typically has the form \(f(x) = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In the given exercise, the function is \(f(x) = 3x\), which is a special type of linear function that passes through the origin with no y-intercept, meaning \(b = 0\). This results in a straight line that captures the essence of linearity with a constant slope, making them predictable and easy to work with.

In the exercise, after finding the inverse, our linear functions \(f(x) = 3x\) and \(f^{-1}(x) = x/3\) have slopes that are reciprocals of each other (3 and 1/3, respectively). Understanding linear functions allows us to easily determine the inverse by swapping the roles of the input and output variables, leading to precise graph manipulations and insights.

Graphing utilities

Graphing utilities are powerful tools for visualizing mathematics, especially when exploring functions and their inverses. They allow us to plot complex equations and instantly see the relationships between them. Using graphing utilities, both \(f(x) = 3x\) and its inverse \(f^{-1}(x) = x/3\) can be drawn in the same coordinate plane, providing a clear visual of their relationship.

By plotting both functions, you'll notice that each is a straight line passing through the origin but with different slopes. With graphing utilities, the difference in angle and steepness becomes apparent, helping us understand how they interact. Another useful feature of graphing utilities is the ability to visually confirm properties, such as symmetry, by observing the reflections of two functions. For students, graphing utilities provide an immediate feedback loop that aids in the understanding and verification of algebraic work.

Symmetry in graphs

Symmetry in graphs is an important concept, often seen when dealing with inverse functions. When you graph a function and its inverse, such as \(f(x) = 3x\) and \(f^{-1}(x) = x/3\), you will observe that their graphs are symmetric with respect to the line \(y = x\). This symmetrical property is a hallmark of inverse functions.

The line \(y = x\) acts like a mirror, and each point on one function corresponds to a reflection onto the other. For instance, if there is a point (a, b) on \(f(x)\), you will find a corresponding point (b, a) on \(f^{-1}(x)\). This characteristic is a direct result of the way inverses are derived by swapping input and output. Understanding symmetry helps to grasp not only the theoretical foundation of inverses but also provides comfort in checking the correctness of inverses. Visible symmetry reaffirmed by graphing adds confidence in the solution and appreciation for the elegance of mathematics.