Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$y=-2 x^{5}-4$$

Step-by-Step Solution

Answer

The function is one-to-one because it is strictly decreasing.

Step 1: Understand the Definition of One-to-One Functions

A function is one-to-one if for every distinct pair of inputs, the outputs are distinct. In other words, if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \). This means that each output is produced by exactly one input.

Step 2: Analyze the Function Type

The given function is \( y = -2x^5 - 4 \). Notice that the function involves an odd-powered polynomial, \( x^5 \), which suggests that the function could be one-to-one. Odd-powered polynomials are typically one-to-one because they consistently either increase or decrease across their domain.

Step 3: Check the Function's Derivative

To ensure the function is one-to-one, we'll verify that the function is either strictly increasing or strictly decreasing everywhere by checking the derivative, \( y' \). Calculate \( y' = \frac{d}{dx}(-2x^5 - 4) = -10x^4 \).

Step 4: Analyze the Derivative

The derivative \( y' = -10x^4 \) is always non-positive since \(-10x^4 \leq 0\) for all real \( x \). Specifically, \( y' = 0 \) only at \( x = 0 \), but around \( x = 0 \), the function remains non-increasing because \( y' \) does not change sign. Hence, the function is strictly decreasing, confirming it is one-to-one.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of one or more variables multiplied by coefficients. For example, the function given in this exercise, \( y = -2x^5 - 4 \), is a polynomial function because it follows this structure. It contains a single term with a power of 5, making it a fifth-degree polynomial.

• The coefficients determine the shape and position of the graph.
• The highest power of the variable (in this case, 5) is called the degree of the polynomial.

Understanding the degree is crucial as it influences the number of turning points and the overall behavior of the function as \( x \) approaches large positive or negative values. Odd-degree polynomials, like this one, tend to have opposite ends of the graph extending towards infinity and negative infinity, respectively, which often leads to the function being one-to-one.

Derivative

In calculus, the derivative of a function measures how the output changes with a change in the input. For polynomial functions, taking the derivative involves applying the power rule, which states that if \( f(x) = ax^n \), the derivative \( f'(x) \) is \( anx^{n-1} \).

For the function \( y = -2x^5 - 4 \), the derivative is calculated as follows:

• Apply the power rule: \( y' = \frac{d}{dx}(-2x^5 - 4) = -10x^4 \).
• This tells us the rate at which \( y \) changes with respect to \( x \).

The derivative being \( -10x^4 \), which is always non-positive, means that the function’s slope is either zero or negative at any point on the graph. This helps determine the behavior of the original function over its domain, specifically in terms of being strictly increasing or decreasing.

Strictly Decreasing Function

A strictly decreasing function is one where the output decreases as the input increases. Here, the derivative \( y' = -10x^4 \) plays a crucial role. Since this derivative is always non-positive (\(-10x^4 \leq 0\)), it indicates that the graph of the function does not rise anywhere, hence it is always going down or remaining flat.

• This aspect is evident here, as the derivative is zero only at \( x = 0 \), but because there is no sign change around zero, the descent is steady.
• Because the function does not increase at any part of its domain, it ensures each output is linked to a unique input.

When a function is strictly decreasing, like this one, it's naturally one-to-one over its entire domain, satisfying the condition that different inputs lead to different outputs. This property is instrumental in confirming that \( y = -2x^5 - 4 \) is indeed a one-to-one function.