Problem 70 - Calculus for Biology and Medicine

Question

(a) Show that \(f(x)=x^{3}-1, x \in \mathbf{R}\), is one to one, and find its inverse together with its domain. (b) Graph \(f(x)\) and \(f^{-1}(x)\) in one coordinate system, together with the line \(y=x\), and convince yourself that the graph of \(f^{-1}(x)\) can be obtained by reflecting the graph of \(f(x)\) about the line \(y=x\).

Step-by-Step Solution

Answer

(a) Yes, \( f(x) = x^3 - 1 \) is one-to-one. The inverse is \( f^{-1}(x) = \sqrt[3]{x + 1} \). (b) Graphs confirm reflection about \( y = x \).

Step 1: Verify One-to-One Function

To show that the function \( f(x) = x^3 - 1 \) is one-to-one, we use the derivative method. Calculate the derivative: \( f'(x) = 3x^2 \). Since \( f'(x) > 0 \) for all \( x eq 0 \), the function is strictly increasing everywhere except at zero, where it doesn't decrease, confirming that \( f(x) \) is one-to-one for \( x \in \mathbb{R} \).

Step 2: Find the Inverse Function

To find the inverse of \( f(x) = x^3 - 1 \), solve the equation \( y = x^3 - 1 \) for \( x \):\[ y + 1 = x^3 \]Take the cube root of both sides: \[ x = \sqrt[3]{y + 1} \].Thus, the inverse function is \( f^{-1}(x) = \sqrt[3]{x + 1} \).

Step 3: Determine the Domain of the Inverse

The original function \( f(x) = x^3 - 1 \) is defined for all real numbers. Hence its range is also all real numbers since as \( x \to \infty \), \( y \to \infty \) and as \( x \to -\infty \), \( y \to -\infty \). Therefore, the domain of \( f^{-1}(x) \) is all real numbers as well, \( x \in \mathbb{R} \).

Step 4: Graph the Functions

Graph the function \( f(x) = x^3 - 1 \), which is a cubic function shifted down by 1 unit, and its inverse \( f^{-1}(x) = \sqrt[3]{x + 1} \), which is the cube root function shifted left by 1 unit, on the same set of axes. Also, draw the line \( y = x \). The graph of \( f^{-1}(x) \) should appear as a reflection of \( f(x) \) across the line \( y=x \).

Key Concepts

One-to-One Functions

A function is considered one-to-one if it assigns a unique output for every distinct input. This means that no two different inputs will produce the same output. Understanding this concept is crucial, especially when it comes to finding inverse functions.

To determine if a function is one-to-one, we often use techniques such as the horizontal line test on a graph or calculus-based methods like evaluating the derivative of the function.

• **Horizontal Line Test**: A function is one-to-one if no horizontal line intersects its graph more than once.
• **Calculus Method**: You can check if the function is strictly increasing or decreasing by examining its derivative. If the derivative is positive or negative everywhere except possibly at isolated points, the function is one-to-one.

In the case of the function \( f(x) = x^3 - 1 \), we used the derivative test. The derivative \( f'(x) = 3x^2 \) is always non-negative and positive for all \( x eq 0 \). Therefore, the function is strictly increasing, indicating that it is one-to-one.

Derivative Test

The derivative test is a method used in calculus to determine whether a function is one-to-one.

By taking the derivative of the function, you can examine its rate of change across its domain. A strictly increasing or decreasing function implies uniqueness of outputs, hence, it will be one-to-one.

For the function \( f(x) = x^3 - 1 \), we took its derivative \( f'(x) = 3x^2 \). Since this derivative is always non-negative and strictly positive for \( x eq 0 \), it confirms the function is always increasing.

• When \( f'(x) > 0 \), the function is strictly increasing.
• When \( f'(x) < 0 \), the function is strictly decreasing.
• When a function is constantly increasing or decreasing across its entire domain, it is one-to-one.

Thus, the derivative test verified that \( f(x) = x^3 - 1 \) is one-to-one.

Graph Reflecting

Graph reflecting is a useful technique in visualizing inverse functions. It involves flipping the graph of a function about the line \( y = x \), which helps in understanding how the inverse function behaves.

When a function \( f(x) \) and its inverse \( f^{-1}(x) \) are plotted on the same coordinate system, their graphs are symmetrical about the line \( y = x \).

• This line \( y = x \) acts as a mirror, where each point on \( f(x) \) has a corresponding reflected point on \( f^{-1}(x) \).
• The reason for this symmetry is the nature of inverse functions to reverse the input-output pairs of the original function.

For our example, when you graph \( f(x) = x^3 - 1 \) and its inverse \( f^{-1}(x) = \sqrt[3]{x + 1} \), you should see that flipping the graph of \( f(x) \) around the line \( y = x \) gives you the graph of \( f^{-1}(x) \).

Domain of a Function

The domain of a function refers to the set of all possible input values. It’s the range of independent variable values for which the function is defined.

For finding the inverse, understanding the domain of the original function is crucial as it becomes the range for the inverse function.

• If \( f(x) \) is defined for all real numbers, then its inverse \( f^{-1}(x) \) must also be defined for all real numbers, since it just flips the domain and range.
• The process of determining the inverse involves solving \( y = f(x) \) for \( x \), often under specific conditions imposed by \( f \). Once solved, you interpret the result to understand the domain of the inverse function.

In the example, the function \( f(x) = x^3 - 1 \) is defined for all real numbers \( x \). Therefore, the range of \( f(x) \) is also all reals, making the domain of \( f^{-1}(x) = \sqrt[3]{x + 1} \) also all real numbers. This global property is key in understanding the full extent of the function's applications.