Problem 20 - A Graphical Approach to College Algebra

Question

Decide whether each function is one-to-one. Do not use a calculator. $$y=-2 x^{5}-4$$

Step-by-Step Solution

Answer

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

Step 1: Understanding One-to-One Functions

A function is considered one-to-one if for every pair of different input values, the corresponding output values are also different. This means that no horizontal line intersects the graph of the function more than once.

Step 2: Check for Monotonicity

Since the given function, \( y = -2x^5 - 4 \), is a polynomial, we can check if it is monotonic (either always increasing or always decreasing) as one way to determine if it's one-to-one. We will do this by examining the derivative.

Step 3: Find the Derivative

Differentiate the function with respect to \( x \) to find \( y' \). For \( y = -2x^5 - 4 \), the derivative \( y' = -10x^4 \).

Step 4: Analyze the Derivative

The derivative \( y' = -10x^4 \) is always non-positive and zero only when \( x = 0 \). For all other values of \( x \), \( y' \) is negative, meaning the function is always decreasing.

Step 5: Conclusion on One-to-One

Since the derivative is negative (aside from point \( x = 0 \)), the function is strictly decreasing everywhere except at a single point. Therefore, \( y = -2x^5 - 4 \) does not repeat any output values for distinct input values, making it a one-to-one function.

Key Concepts

Polynomial Functions

Polynomial functions, like the one given in the original exercise, involve terms that are powers of a variable combined with coefficients. The function in question is: \[ y = -2x^5 - 4 \].
Here, the highest power of the variable \( x \) determines the degree of the polynomial, which is 5 in this case. Generally, higher degree polynomials have more complex graphs with multiple turning points.

• Odd-degree polynomials, such as this one, can have behaviors that increase to infinity and decrease to negative infinity.
• The leading coefficient and the power of the highest term together determine the end behavior of the function. Here, \(-2x^5\) dictates that as \( x \) increases or decreases, the outputs become increasingly negative.

Understanding these characteristics is crucial when analyzing functions for properties like being one-to-one.

Derivative

The derivative of a polynomial function is a key tool in determining the characteristics of its graph. In this case, the function is \( y = -2x^5 - 4 \), and we compute the derivative:

• Applying basic differentiation rules, we find \( y' = -10x^4 \).
• This tells us about the rate of change of the function at any given point.
• In essence, the derivative function gives us a way to visualize how the original function behaves.

This is crucial because the sign of the derivative helps us understand whether the function is increasing or decreasing.

Monotonicity

Monotonicity refers to whether a function is consistently increasing or decreasing. For the polynomial\( y = -2x^5 - 4 \), its derivative is \( y' = -10x^4 \). Since the derivative is:

• Non-positive for all \( x \) (i.e., \( y' \leq 0 \)),
• Zero only at \( x = 0 \),

It indicates the function is always decreasing outside of where the derivative is zero. This consistent behavior implies that the function does not increase at any point, strengthening the case that it is monotonic. A monotonic function helps to simplify the process of determining one-to-oneness.

Function Analysis

The thorough analysis of a function like \( y = -2x^5 - 4 \) involves several steps. Assessing its derivative, monotonicity, and overall behavior help us conclude its nature. Here's how the analysis unfolds:

• The negative derivative confirms the function is always decreasing, except at a single point where it is constant.
• This indicates that each unique input results in a unique output – the hallmark of a one-to-one function.
• By checking these aspects, we conclusively decide if any output value is repeated for distinct inputs.

Thus, analyzing these features lets us determine whether the function meets the criteria as a one-to-one function.