Problem 34 - Precalculus : Building Concepts and Connections

Question

State whether each function is one-to-one. $$f(x)=-\frac{1}{3} x^{3}-5$$

Step-by-Step Solution

Answer

Yes, the function \(f(x)=-\frac{1}{3} x^{3}-5\) is one-to-one.

Step 1: Differentiate the function

The first step is to find the derivative of the function. The derivative of a function tells us if the function is increasing or decreasing. If the derivative is positive, the function is increasing, and if it's negative, the function is decreasing. The derivative of the function \(f(x)=-\frac{1}{3} x^{3}-5\) is \(f'(x)=-x^{2}\).

Step 2: Analyze the derivative

Now, let's determine where this derivative is positive and where it's negative. Since \(f'(x)=-x^{2}\), we can see that the derivative will be negative for any real number \(x\). Since the derivative never changes sign, this means that the original function is decreasing over its entire domain.

Step 3: Conclude if the function is one-to-one

Since the original function \(f(x)\) is decreasing over its entire domain, that means that for every input \(x\), there's a unique output. This is precisely the characteristic of a one-to-one function. Therefore, the function \(f(x)=-\frac{1}{3} x^{3}-5\) is one-to-one.

Key Concepts

Derivatives

In mathematics, derivatives are crucial in understanding the behavior of functions. A derivative represents the rate at which a function is changing at any point. For example, when you compute the derivative of a function, you're identifying its slope at each point. This slope tells us whether a function is increasing or decreasing.

Let's consider the function given in the exercise: \(f(x) = -\frac{1}{3}x^3 - 5\). When you differentiate this function, you get \(f'(x) = -x^2\). Derivatives help us capture instantaneous rates of change and are vital tools in calculus for analyzing the behavior of functions.

• A positive derivative indicates an increasing function.
• A negative derivative suggests a decreasing function.
• A zero derivative implies stationary points like peaks or valleys.

This derivative, \(-x^2\), is always negative, confirming the function's decreasing nature over all real numbers.

Decreasing Functions

A function is described as decreasing if as the input increases, the output decreases. This means that for any two numbers, say \(x_1 < x_2\), \(f(x_1) > f(x_2)\). Decreasing functions display a downward trend when graphed.

With the function \(f(x) = -\frac{1}{3}x^3 - 5\), its derivative \(-x^2\) is negative for all values of \(x\), indicating that it is decreasing across its domain. Essentially, a function without intervals of increasing segments is deemed decreasing entirely.

• Always look at the sign of the derivative to determine whether it is decreasing.
• A consistently negative derivative confirms a function's consistently decreasing nature.
• In practical terms, this kind of function is one that never reverses direction upwards.

Understanding decreasing functions is crucial for informing decisions on one-to-one characteristics of functions.

Function Analysis

Function analysis involves a variety of techniques used to understand the behavior and characteristics of functions, such as determining their one-to-one nature. Functions can be analyzed for increasing or decreasing trends, extreme values, continuity, and many more properties.

For the function \(f(x) = -\frac{1}{3}x^3 - 5\), our function analysis reveals it is a one-to-one function because it is entirely decreasing. This means every value of \(x\) yields a distinct \(f(x)\), a defining property of one-to-one functions usage.

• One-to-one functions map each element of a domain to a unique range.
• They are important in inverses, as only one-to-one functions have inverses that are also functions.
• Analyzing derivatives helps identify these functions, ensuring no repeated values in outputs for different inputs.

Through careful function analysis, you can draw meaningful conclusions about a function's behavior and properties.