Problem 27 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=x^{4} $$

Step-by-Step Solution

Answer

The function \(f(x) = x^4\) is not one-to-one.

Step 1: Understand One-to-One Functions

A function is considered one-to-one if each output or 'y' value is paired with only one input or 'x' value. This means that no horizontal line intersects the graph of the function more than once.

Step 2: Apply the Horizontal Line Test

A function is one-to-one if no horizontal line intersects its graph more than once. For the function \(y = x^4\), consider its graph: \(x^4\) is an even function, symmetric about the y-axis, opening upwards like a parabola.

Step 3: Graph Analysis

The graph of \(f(x) = x^4\) is a parabola opening upwards with its minimum at the origin. Visually, any horizontal line above the x-axis will intersect the graph at two points, except at the vertex.

Step 4: Conclusion Based on the Horizontal Line Test

Since horizontal lines intersect \(f(x) = x^4\) at more than one point above the x-axis, the function fails the horizontal line test. Therefore, it is not one-to-one.

Key Concepts

Graphical Interpretation

When we analyze the function graphically, it becomes easier to visually determine certain properties of the function. The function given is \( f(x) = x^4 \), which exhibits particular graphical characteristics. This function resembles the shape of a parabola, specifically one that opens upwards. Unlike the basic parabola of \( f(x) = x^2 \), the graph of \( x^4 \) is flatter near the origin and becomes steeper as \( x \) moves away from zero. This is because the degree of \( x \) is higher in \( x^4 \) compared to \( x^2 \), increasing its growth rate.

In general, when you have a polynomial function where the highest degree term has an even exponent, the graph tends to show mirror symmetry about the y-axis. This symmetry is a crucial hint towards understanding one of the core qualities of the graph: being an even function. Appreciating the symmetry in the graph can help us anticipate how various tests, like the horizontal line test, will behave. It guides the analysis of whether a function could potentially be one-to-one.

Horizontal Line Test

The horizontal line test is a simple yet powerful tool in determining if a function is one-to-one. The essence of this test is straightforward: a function is considered one-to-one if no horizontal line can intersect the graph of the function at more than one point. Here’s what happens when you apply this test to \( f(x) = x^4 \):

The graph of \( x^4 \) is symmetric about the y-axis. As you draw horizontal lines above the x-axis, you'll notice that these lines often intersect the graph at two points, except exactly at the minimum point (the vertex) of the parabola. This gives us an immediate visual cue that \( x^4 \) does not pass the horizontal line test.

• The exception is the line passing through the origin, which touches the vertex once.
• For any other horizontal line, multiple intersections exist.

As each horizontal line typically intersects the curve more than once, we conclude that this function is not one-to-one.

Even Function

Understanding the nature of even functions adds another layer to interpreting their graphs. A function is considered even if \( f(-x) = f(x) \) for all \( x \) in its domain. This mathematical symmetry implies that the function’s graph is mirrored across the y-axis.

Despite providing a clear and organized structure, even functions often struggle with the concept of being one-to-one. In the case of \( x^4 \), both \( x \) and \( -x \) yield the same output. For example, \( f(2) = 16 \) and \( f(-2) = 16 \). This symmetry fulfills the criteria of being even but fails the criteria of being one-to-one.

• This mirrored effect inherently suggests that several \( x \) values share the same \( y \) value.
• Such properties affirm that \( x^4 \) is not one-to-one, as multiple inputs lead to identical outputs.

Recognizing this symmetry helps us see that many even functions, like \( x^4 \), fail the horizontal line test due to their structural characteristics.