Problem 21 - Elementary and Intermediate Algebra

Question

Determine whether each function is one-to-one. \(f(x)=x^{4}\)

Step-by-Step Solution

Answer

The function \( f(x) = x^4 \) is not one-to-one because it does not pass the horizontal line test.

Step 1: Understand One-to-One Function Definition

A function is one-to-one if every element of the range is mapped to by exactly one element in its domain. This means that if \(f(a) = f(b)\), then \(a = b\). In simpler terms, a one-to-one function passes the horizontal line test: no horizontal line intersects the graph of the function at more than one point.

Step 2: Set the Function Equal for Two Different Variables

We need to check whether \( f(a) = f(b) \) implies \( a = b \). For the function \( f(x) = x^4 \), let's assume \( f(a) = f(b) \). So, \( a^4 = b^4 \).

Step 3: Simplify the Equation

To simplify \( a^4 = b^4 \), take the fourth root of both sides. This results in two possibilities: \( a = b \) or \( a = -b \). This implies that more than one \(a\) can map to the same \(b\), breaking the one-to-one property.

Step 4: Perform the Horizontal Line Test Visually

Graphically, the function \( f(x) = x^4 \) is symmetric with respect to the y-axis and resembles a parabolic shape. Any horizontal line across the y-axis will intersect the graph at more than one point, indicating it is not one-to-one.

Key Concepts

Horizontal Line Test

The horizontal line test is a handy graphical tool used to determine if a function is one-to-one. A function is one-to-one if and only if no horizontal line cuts across its graph more than once.
Here's how it works:

• Draw horizontal lines across your graph, spanning the y-axis.
• If any of these lines intersect the graph at multiple points, the function fails the test and is not one-to-one.
• Conversely, if every horizontal line only touches the graph at a single point, the function is one-to-one.

Applying this test to the function \( f(x) = x^4 \), we notice that lines parallel to the x-axis can cross it at multiple points. This means \( f(x) = x^4 \) is not a one-to-one function. The test shows that for values like \( x = 2 \) and \( x = -2 \), \( f(x) = 16 \), leading to more than one input producing the same output.

Function Symmetry

Function symmetry tells us about the evenness or oddness of a function, indicating whether it mirrors itself across an axis or origin.
For \( f(x) = x^4 \):

• This function is symmetric about the y-axis because \( f(x) = f(-x) \). So, \( (3)^4 = (-3)^4 \), resulting in the same output of 81.
• Such symmetry is a characteristic of even functions, making them not one-to-one by default due to multiple domain elements resulting in the same range values.

The symmetry of \( f(x) = x^4 \) reveals why it isn't one-to-one. As it's symmetric with respect to the y-axis, similar positive and negative values provide identical outputs, failing to adhere to the one-to-one condition.

Fourth Root

The concept of the fourth root involves undoing an exponent of four. In simpler terms, solving a problem involving the fourth root is akin to asking, "What number, multiplied by itself four times, gives this result?"
When dealing with equations like \( a^4 = b^4 \) and extracting their roots:

• Taking the fourth root of both sides gives \( a = b \) or \( a = -b \).
• This results in two possible solutions, showing that different numbers can lead to the same outcome when raised to the fourth power.

The fourth root in this context explains why \( f(x) = x^4 \) is not one-to-one. With both \( a = 2 \) and \( a = -2 \) returning the same fourth power value, it becomes evident that multiple domain values connect to a single range value, violating the strict requirements for a one-to-one function.