Problem 10 - Algebra for College Students

Question

Determine whether the function \(f\) is one-to-one. $$ f(x)=x^{5}+1 $$

Step-by-Step Solution

Answer

The function \( f(x) = x^5 + 1 \) is one-to-one.

Step 1: Understand One-to-One Function

A function is one-to-one if and only if for every pair of different inputs, the outputs are different. This means if \( f(x_1) = f(x_2) \), then it must follow that \( x_1 = x_2 \). We need to show this property for the given function \( f(x) = x^5 + 1 \).

Step 2: Consider the Function's Form and Inverse Nature

We first recognize that \( f(x) \) is a polynomial of odd degree (5th degree). Polynomials of odd degrees that are strictly increasing or decreasing by nature generally tend to be one-to-one. For a more detailed check, we can confirm by using the derivative to analyze the monotonicity of the function.

Step 3: Find the Derivative of the Function

To use calculus to help determine if \( f(x) \) is one-to-one, find the derivative of \( f(x) \). The derivative \( f'(x) \) is computed as follows: \[ f'(x) = \frac{d}{dx}(x^5 + 1) = 5x^4. \]

Step 4: Analyze the Derivative for Monotonicity

The derivative \( f'(x) = 5x^4 \) is always non-negative since \( x^4 \) is non-negative for all real \( x \), making \( 5x^4 \geq 0 \). It equals zero only at \( x=0 \), but because the general nature of \( x^4 \) implies local behavior around zero that does not repeat outputs, \( f(x) \) remains strictly increasing (in a broad sense) elsewhere.

Step 5: Conclusion

Since the derivative \( f'(x) = 5x^4 \geq 0 \) and is zero only at a single point (which does not change the overall increasing nature around it), the function \( f(x) \) is strictly increasing on its entire domain. This implies \( f(x) \) is a one-to-one function because there are no multiple outputs for a given input across the whole range of real numbers.

Key Concepts

Polynomial Functions

Polynomial functions form the backbone of algebra and calculus. They are expressed as sums of terms consisting of a variable raised to a whole number power, multiplied by a coefficient. For example, the function \( f(x) = x^5 + 1 \) is a polynomial function.

• Polynomials can have various degrees, which are determined by the highest power of \( x \).
• The degree of a polynomial significantly influences its shape and behavior.
• Polynomials are continuous and smooth functions, meaning they have no breaks or holes.

Recognizing that \( x^5 \) is the highest degree term in our function, we observe that it is an odd-degree polynomial. Odd-degree polynomials, unlike even-degree, tend to stretch infinitely in both directions, ensuring there are no horizontal asymptotes or repeating values across infinite domains.

Function Monotonicity

Monotonicity is a vital characteristic of a function in determining whether it is one-to-one. When a function is monotonic, it either consistently increases or decreases over its entire domain.

• A monotonic increasing function continually rises as the input \( x \) increases.
• A monotonic decreasing function continually drops as \( x \) increases.
• If a function is strictly monotonic, it implies that it is one-to-one over its domain.

For the polynomial function \( f(x) = x^5 + 1 \), examining its derivative helps us understand its monotonicity. The function \( f(x) \) is strictly increasing because its derivative is non-negative throughout the entire real line.

Calculus Derivatives

Derivatives play a crucial role in understanding functions, particularly in examining their behavior and characteristics like monotonicity. The derivative of a function essentially gives us the rate at which the function's output changes concerning its input.For the function \( f(x) = x^5 + 1 \), the derivative is calculated as follows:\[ f'(x) = \frac{d}{dx}(x^5 + 1) = 5x^4\]

• The derivative \( f'(x) = 5x^4 \) gives us insight into whether the graph of \( f \) is rising or falling.
• The expression \( 5x^4 \) is always non-negative, indicating that \( f(x) \) is non-decreasing.
• The derivative is zero only at \( x = 0 \), suggesting a flat point rather than a change in direction.

This confirms the function is strictly increasing everywhere but tends to remain constant only briefly at \( x = 0 \). Therefore, \( f(x) = x^5 + 1 \) is one-to-one, as its derivative supports consistent monotonicity across its domain.