Problem 47 - Single Variable Calculus

Question

Let g denote the inverse of the function \(f\). (a) Show that the point \((a, b)\) lies on the graph of \(f .\) (b) Find \(g^{\prime}(b) .\) Suppose that \(g\) is the inverse of a function \(f .\) If \(f(2)=4\) and \(f^{\prime}(2)=3\), find \(g^{\prime}(4) .\)

Step-by-Step Solution

Answer

The point \((a, b)\) lies on the graph of \(f\) since \(f(a) = b\). The derivative of the inverse function \(g'(b)\) is given by the formula \(g'(b) = \frac{1}{f'(a)}\). Given that \(f(2) = 4\) and \(f'(2) = 3\), the derivative of the inverse function at \(g'(4)\) is \(\frac{1}{3}\).

Step 1: Prove that point \((a, b)\) lies on the graph of \(f\)

Recall that the inverse function \(g\) is defined as \(g(f(x)) = x\). So, to show that the point \((a, b)\) lies on the graph of \(f\), we need to show that \(f(a) = b\). Since \(g\) is the inverse function of \(f\), we have \(g(f(a)) = a\). Then, we can replace the given point into the equation: \(g(b) = a\). Now, we can apply the definition of an inverse function again, and this time with \(g(b) = a\), giving us \(f(a) = b\). Thus, the point \((a, b)\) lies on the graph of \(f\).

Step 2: Find the derivative of \(g\), \(g'(b)\)

To find the derivative of the inverse function \(g'(b)\), we can use the formula for the derivative of an inverse function: \(g'(b) = \frac{1}{f'(a)}\), where \(f(a) = b\).

Step 3: Use given values for \(f\) and \(f'\) to find \(g'(4)\)

We are given that \(f(2) = 4\) and \(f'(2) = 3\). We need to find \(g'(4)\). First, we can observe that \(f(2) = 4\), which indicates that \((2, 4)\) lies on the graph of \(f\). Therefore, \((4, 2)\) lies on the graph of the inverse function \(g\), since \(g(4) = 2\). Now, we can use the formula for the derivative of an inverse function: \(g'(b) = \frac{1}{f'(a)}\), where \(f(a) = b\). In our case, we have \(a = 2\), \(b = 4\), and \(f'(a) = f'(2) = 3\). Plug these values into the formula: $$g'(4) = \frac{1}{f'(2)} = \frac{1}{3}$$ So, the derivative of the inverse function \(g'(4)\) is \(\frac{1}{3}\).

Key Concepts

Definition of Inverse Function

Understanding the concept of an inverse function is crucial when we explore the relationship between two interdependent sets of data. In mathematics, an inverse function reverses the effect of the original function. Put simply, if you have a function that takes an input and provides an output, the inverse function takes that output and recovers the original input.

For a function \( f(x) \) to have an inverse, it must be bijective, which means it is both injective (one-to-one) and surjective (onto). The notation for an inverse function is usually written as \( f^{-1}(x) \) or \( g(x) \) if we are naming the original function \( f(x) \). It is important to remember that not all functions have inverses, specifically those that are not one-to-one. Graphically, the inverse is reflected over the line \( y = x \) when you plot the original function and its inverse on the same axes.

Derivative of Inverse Function

The concept of the derivative of an inverse function might sound complex at first, but it follows a logical progression from the fundamentals of calculus. The derivative tells us the rate at which one quantity changes with respect to another. When dealing with inverse functions, we use a special rule to find this rate of change.

The formula for the derivative of an inverse function is given by \( g'(b) = \frac{1}{f'(a)} \), where \( f(a) = b \) and \( g \) is the inverse of \( f \). This formula derives from the chain rule in calculus and essentially states that the slope of the tangent line at a point on the graph of the inverse function is the reciprocal of the slope of the tangent line at the corresponding point on the graph of the original function.

Derivative Calculation

The process of derivative calculation often strikes fear into the hearts of students, but it doesn't have to be intimidating. The derivative represents the instantaneous rate of change of a function with respect to its variable. It's the slope of the curve of the function at any given point. Practically, you can think of it as the 'speedometer' reading of a function's graph at a specific moment.

To calculate the derivative, we use several rules and formulas, including the power rule, product rule, quotient rule, and chain rule. To find the derivative of an inverse function, we apply the formula mentioned earlier, making use of the original function's derivative. In many practical examples, you will be given specific values for the function and its derivative at certain points, from which you can determine the derivative of its inverse.

Graph of a Function

A graph of a function is a visual representation of all the possible points \( (x, f(x)) \) that lie on the Cartesian plane. A well-graphed function can offer tremendous insight into the behavior of the function over its domain, such as identifying maxima, minima, and points of inflection.

To graph a function, you typically plot points for various values of \( x \) and then draw a curve through these points. When dealing with inverse functions, their graphs are mirrors of each other over the line \( y = x \). This mirroring occurs because each \( (x, y) \) point on the original function corresponds to a \( (y, x) \) point on the inverse function. As you graph more complex functions and their inverses, the symmetrical relationship between the two becomes an essential tool for understanding how they relate to each other.