Problem 66 - 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 one-to-one because it is strictly increasing.

Step 1: Understand the function

The given function is \( f(x) = x^3 + x \). This is a polynomial function of degree 3, which means it is continuous and smooth with no breaks or jumps.

Step 2: Identify the basic behavior

Because this is a cubic function (higher odd degree), we expect its graph to be an S-shaped curve starting at the bottom left and going to the top right.

Step 3: Find critical points

Calculate the derivative of the function, \( f'(x) = 3x^2 + 1 \). The derivative can never be zero because \( 3x^2 + 1 \) is always positive for all \( x \).

Step 4: Analyze monotonicity

Since the derivative \( f'(x) = 3x^2 + 1 \) is always positive, the function \( f(x) = x^3 + x \) is strictly increasing on its entire domain, \( (-\infty, \infty) \).

Step 5: Draw the graph

Sketch a graph that starts from the bottom left, moves smoothly upwards, and passes through the origin. It should curve upwards continuously since it is strictly increasing.

Step 6: Determine if function is one-to-one

A function is one-to-one if it never takes the same value twice. Since the function is strictly increasing (as shown by the derivative always being positive), it is indeed one-to-one.

Key Concepts

One-to-One Functions

When we talk about one-to-one functions, we're referring to functions that assign each input exactly one unique output. This means that no horizontal line crosses the graph of the function more than once. In mathematical terms, a function is one-to-one if, for any two different inputs, the outputs are also different.
To determine if a function is one-to-one, we often rely on the derivative. For the function \( f(x) = x^3 + x \):

• We found its derivative as \( f'(x) = 3x^2 + 1 \).
• This derivative is always positive (since \( 3x^2 + 1 \geq 1 \) for all \( x \)), indicating that the function is always increasing.

This constant increase means the function never repeats a value, confirming that \( f(x) \) is one-to-one. Understanding one-to-one functions is crucial for considering function inverses, as only one-to-one functions have inverses that are also functions.

Graphing Functions

Graphing is a powerful way to visualize functions, allowing us to understand their behaviors better. When we graph \( f(x) = x^3 + x \), we focus on several key characteristics:

• The function is a cubic polynomial, known for its S-shaped curve.
• The graph will pass through the origin because substituting \( x = 0 \) gives \( f(0) = 0^3 + 0 = 0 \).

As we sketch this function:

• The continuous, smooth curve suggests no jumps or breaks.
• It starts from the lower left quadrant on the graph and progresses to the upper right, showing its strictly increasing nature.

Graphing is the art of converting numerical and algebraic information into a visual form, helping us immediately notice the characteristics such as growth patterns, symmetry, and whether a function is one-to-one.

Polynomial Functions

Polynomial functions are foundational in algebra, consisting of variables raised to any whole number powers and combined using addition or subtraction. In our example, \( f(x) = x^3 + x \) is a polynomial of degree 3, where:

• The highest power of \( x \) is 3, making it a cubic function.
• Cubic polynomials are especially interesting because they can resemble an S-shape around certain points.

The degree of the polynomial tells us about its end behavior:

• Odd-degree polynomials like \( x^3 \) have ends that move in opposite directions.
• The coefficient of the highest degree term, here implied as \( +1 \), informs us about the direction of this movement. Positive coefficients result in the graph moving upwards to the right.

Understanding polynomial functions also involves recognizing their smoothness and continuity, as well as how they expand or compress based on coefficients. These features make them very predictable and useful for modeling real-world scenarios.