Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=x^{3}+3 x^{2}+6 x+12\). (a) Make a convincing argument that \(f(x)\) is invertible. (It is not adequate to say it looks 1-to-1 on a calculator. How can you be absolutely sure it is 1 -to- 1 on \((-\infty, \infty) ?)\) (b) Find three points that lie on the graph of \(f^{-1}(x)\). (Approximations are not adequate.) Explain your reasoning.

Step-by-Step Solution

Answer

The function \(f(x)\) is invertible as it is strictly increasing and the points (12, 0), (13, -1) and (16, -2) lie on its inverse function \(f^{-1}(x)\).

Step 1: Prove Function Invertibility

To prove that the function \(f(x) = x^{3}+3 x^{2}+6 x+12\) is invertible, take the derivative of \(f(x)\) which is \(f'(x) = 3x^{2} + 6x + 6\). As \(f'(x)\) is always positive for all x, \(f(x)\) is strictly increasing and hence 1-to-1. Therefore, it is invertible on \((-∞, ∞)\).

Step 2: Find Points on the Graph of \(f^{-1}(x)\)

To find the points on \(f^{-1}(x)\), find some points on \(f(x)\) and switch the x and y coordinates. The points (0,12), (-1,13) and (-2,16) lie on \(f(x)\) and hence the points (12, 0), (13, -1) and (16, -2) will lie on \(f^{-1}(x)\).

Key Concepts

Derivatives

When discussing the concept of derivatives, it's crucial to understand that a derivative represents the rate of change of a function. More technically, the derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point. For example, if we have a function such as \( f(x) = x^{3}+3x^{2}+6x+12 \), its derivative \( f'(x) \) is calculated using the power rule.

For our function, the derivative is \( f'(x) = 3x^{2} + 6x + 6 \). This derivative plays a key role in determining the function's behavior, such as increasing or decreasing trends. It is the cornerstone of calculus and is fundamental in understanding phenomena in physics, economics, and more – wherever rates at which things change are important.

Increasing Function

An increasing function is one where, as we move along the graph from left to right, the output values rise. This means that for any two points \( a \) and \( b \) where \( a < b \), the function satisfies \( f(a) \leq f(b) \) for a weakly increasing function or \( f(a) < f(b) \) for a strictly increasing function.

The function \( f(x) \) in our exercise is a model example of a strictly increasing function. By proving that its derivative \( f'(x) = 3x^{2} + 6x + 6 \) is always positive, we confirm that the graph consistently ascends as we move along the x-axis. This fact is not just a check in an exercise; it represents a fundamental aspect of how the function behaves and guarantees the function does not 'turn back on itself', important for its invertibility.

One-to-One Function (Injective)

Being a one-to-one function, or injective, means that each element of the range is mapped to by exactly one element of the domain. To put it simply, no two different inputs give the same output. This property is necessary for a function to have an inverse.

For the function in our exercise, the derivative test provides a strong argument for injectivity. Since \( f'(x) > 0 \) for all x, we're ensured that each x produces a unique y, satisfying the condition for a function to be one-to-one. This concept is critical when considering functions in various fields of study, particularly in mathematics when dealing with the concept of inverse functions.

Inverse Functions

Inverse functions are a pair to their original functions in a way that one 'undoes' the effect of the other. If we have a function \( f(x) \), its inverse, denoted as \( f^{-1}(x) \), fulfills the condition that \( f^{-1}(f(x)) = x \) for all x in the domain of \( f \).

In the context of our example, the points on the graph of \( f(x) \), such as (0,12), can be translated into points on the graph of \( f^{-1}(x) \) by flipping the x and y coordinates to (12,0). Finding these points is not just about graphing but understanding how the output of a function can relate back to its input. The technique of finding an inverse function requires the original function to be one-to-one, connecting back to the previous concept.