Problem 45 - Thomas Calculus in SI Units

Question

a. Find the inverse of the function \(f(x)=m x,\) where \(m\) is a constant different from zero. b. What can you conclude about the inverse of a function \(y=f(x)\) whose graph is a line through the origin with a nonzero slope \(m ?\)

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = mx\) is \(f^{-1}(x) = \frac{x}{m}\). The inverse of a line through the origin with slope \(m\) is a line through the origin with slope \(\frac{1}{m}\).

Step 1: Understanding the Inverse Function

To find the inverse of the function \(f(x) = m x\), we need to find a function \(f^{-1}(x)\) such that when composed with \(f(x)\), it yields the identity function: \(f(f^{-1}(x)) = x\).

Step 2: Expressing the Function in Terms of y

Start by expressing the function \(f(x) = m x\) as \(y = m x\). The goal is to solve for \(x\) in terms of \(y\) to find the inverse function.

Step 3: Solving for x

Solve the equation \(y = m x\) for \(x\). Divide both sides by \(m\), giving us \(x = \frac{y}{m}\).

Step 4: Determining the Inverse Function

Replace \(y\) with \(x\) to express the inverse function. So, the inverse function is \(f^{-1}(x) = \frac{x}{m}\).

Step 5: Concluding About the Graph

For any linear function of the form \(y = mx\) (a line through the origin with slope \(m\)), its inverse \(y = \frac{x}{m}\) is also a line through the origin but with slope \(\frac{1}{m}\). Therefore, the inverse of a linear function is another linear function.

Key Concepts

Linear Functions

A linear function is a fundamental concept in mathematics that describes functions of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. It's called "linear" because its graph is a straight line. One key feature of linear functions is their simplicity. They're completely characterized by their slope \(m\) and y-intercept \(b\). The slope tells us how steep the line is, while the y-intercept indicates where the line crosses the y-axis.
Linear functions are easy to work with because they follow a constant rate of change. In the exercise, the function \(f(x) = mx\) is a special linear function with \(b = 0\), meaning it passes through the origin \((0,0)\). This specific condition simplifies understanding their inverses.

Identity Function

The identity function is a simple yet important concept in mathematics. It is the function that always returns the input value unchanged. It is written as \(f(x) = x\). When we talk about finding the inverse of a function, we aim to find another function that, when composed with the original, results in the identity function. This means \(f(f^{-1}(x)) = x\).
In the exercise, you found an inverse for the function \(f(x) = mx\). When \(f\) and its inverse \(f^{-1}(x) = \frac{x}{m}\) are composed, the result is \(f(f^{-1}(x)) = m\left(\frac{x}{m}\right) = x\), effectively returning us to the identity function. Thus, the composition of a function and its inverse is always the identity function.

Function Composition

Function composition is combining two functions to produce a new function. Given functions \(f\) and \(g\), their composition is denoted as \((f \circ g)(x) = f(g(x))\).
This concept is key when dealing with inverses because the inverse function is precisely the function that "undoes" the work of the original function. That's why we require that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). In the problem, you calculated \(f(f^{-1}(x))\) for \(f(x) = mx\) and saw how composition led to the identity function, proving correctness of the inverse \(f^{-1}(x) = \frac{x}{m}\). This illustrates the power and necessity of function composition when working with inverses.

Slope of a Line

The slope of a line is an essential aspect of understanding linear functions. Slope, often denoted by \(m\), represents how steep a line is. It's calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. Mathematically, \(m = \frac{\Delta y}{\Delta x}\).
In the context of the exercise, the function \(f(x) = mx\) has a slope \(m\). When we determine its inverse, \(f^{-1}(x) = \frac{x}{m}\), the slope of this inverse function changes to \(\frac{1}{m}\). This is a profound relationship because it shows that the inverse of a linear function is a reflection with regards to the line \(y = x\). Thus, the slope plays a critical role in defining the shape and orientation of both a linear function and its inverse.