Problem 65 - Algebra and Trigonometry

Question

Draw the graph of \(f\) and use it to determine whether the function is one-to- one. $$ f(x)=x^{3}-x $$

Step-by-Step Solution

Answer

The function is not one-to-one; it fails the horizontal line test.

Step 1: Analyze the Function

The function given is \(f(x) = x^3 - x\). Notice, this is a cubic polynomial, where the highest degree is 3, indicating it's typically a snake-shaped curve passing through the origin.

Step 2: Determine the Symmetry

Check for symmetry of the function. Since all exponents of \(x\) in \(f(x) = x^3 - x\) are odd, the function is an odd function. Odd functions are symmetric with respect to the origin.

Step 3: Find the Derivative

Calculate the derivative to find where the function is increasing or decreasing. \(f'(x) = 3x^2 - 1\). Analyze \(f'(x)\) to understand the graph's behavior.

Step 4: Critical Points Analysis

To find critical points, set \(f'(x) = 0\). So, solve \(3x^2 - 1 = 0\) which gives \(x = \pm \sqrt{\frac{1}{3}}\). These points will help determine intervals of increase and decrease.

Step 5: Test Intervals

Evaluate \(f'(x)\) on the intervals \((-\infty, -\sqrt{\frac{1}{3}})\), \((-\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}})\), and \((\sqrt{\frac{1}{3}}, \infty)\). For these intervals, \(f'(x)\) is positive, negative, and positive, respectively. Thus, the function first increases, then decreases, and finally increases.

Step 6: Sketch the Graph

Plot the cubic function using the behavior determined. The graph will show a curve passing through the origin, with local maxima and minima aligning with previous analysis. Observing the graph shows it does not pass the horizontal line test (there is more than one x-value for the same y-value in some regions).

Step 7: Determine One-to-One Behavior

Since the function fails the horizontal line test (as it is not strictly increasing or strictly decreasing), \(f(x) = x^3 - x\) is not one-to-one.

Key Concepts

Cubic Function

A cubic function is a type of polynomial function that has a degree of three. This means the highest power of the variable, typically x, is three, making the general form of a cubic function look like:

• \( f(x) = ax^3 + bx^2 + cx + d \)

Here, the coefficients \( a, b, c, \) and \( d \) can be any real number, but the key is that \( a eq 0 \), which ensures that the function is cubic due to the presence of the \( x^3 \) term.
Cubic functions usually feature one or two "turning points," which are places where the graph changes direction. When graphing, these are typically "snake-shaped," passing through the origin if there is no constant term affecting where the curve crosses the y-axis.
In the case of our function, \( f(x) = x^3 - x \), it has neither a quadratic nor linear independent term, ensuring it has a clear symmetric shape centered around the origin.

Derivative

The derivative of a function helps us understand how the function is changing at any given point and is essential for analyzing the function's graph. The derivative can show where a function is increasing or decreasing.

• For our specific function \( f(x) = x^3 - x \), the derivative is calculated like this: \( f'(x) = 3x^2 - 1 \).

This derivative, \( f'(x) = 3x^2 - 1 \), tells us the rate of change of the function. When the derivative is positive, \( f(x) \) is increasing; when it is negative, the function is decreasing.
To assess whether the function \( x^3 - x \) is ever briefly level, you look at where the derivative equals zero. Solving \( 3x^2 - 1 = 0 \) tackles this.

Critical Points

Critical points of a function are points where the derivative is zero or undefined, marking potential maxima, minima, or points of inflection. These help identify intervals of increase and decrease in the function. To find critical points of \( f(x) = x^3 - x \), we solved:

• \( 3x^2 - 1 = 0 \)
• The solutions are \( x = \pm \sqrt{\frac{1}{3}} \)

These points are crucial in analyzing the behavior of the cubic function:

• \( x = -\sqrt{\frac{1}{3}} \) indicates where the function starts to decrease before reaching a local minimum.
• \( x = \sqrt{\frac{1}{3}} \) marks the shift upwards, defining a local maximum.

This entire behavior ultimately affects the overall one-to-one nature of the function.

Horizontal Line Test

The horizontal line test is a visual way to determine if a function is one-to-one. For a function to be one-to-one, any horizontal line drawn across the graph should intersect it at most once. This ensures that each y-value corresponds to exactly one x-value.
When we applied this test to the function in question, \( f(x) = x^3 - x \), it failed. The graph had sections where horizontal lines intersected the curve at more than one point, indicating multiple x-values for the same y-value.

• Thus, due to these overlaps in the cubic function, \( f(x) \) cannot be considered one-to-one.

This analysis highlights the graphical characteristics leading to the conclusion about the function's one-to-one status.