Problem 21 - 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.

Step 1: Understanding One-to-One Function

A function is considered one-to-one if every output corresponds to exactly one input. In mathematical terms, a function is one-to-one if for all pairs of inputs \(x_1\) and \(x_2\), whenever \(f(x_1) = f(x_2)\) implies that \(x_1 = x_2\). We need to check if this condition holds for \(f(x) = x^4\).

Step 2: Testing the Function Property

Consider two arbitrary inputs \(x_1\) and \(x_2\). We need to investigate the situation \(f(x_1) = f(x_2)\), which implies \(x_1^4 = x_2^4\). Is the only possible conclusion \(x_1 = x_2\)?

Step 3: Analyzing Equality of Powers

Given \(x_1^4 = x_2^4\), we must determine whether \(x_1 = x_2\) with certainty. Notice that \(x_1^2 = x_2^2\), which means either \(x_1 = x_2\) or \(x_1 = -x_2\).

Step 4: Conclusion on One-to-One Property

Since \(x_1^4 = x_2^4\) leads not only to \(x_1 = x_2\) but also to \(x_1 = -x_2\), the function is not one-to-one. Multiple inputs (e.g., 2 and -2) yield the same output (16).

Key Concepts

Function Analysis

Function analysis involves investigating the properties and behavior of a function, such as its domain, range, and whether it is one-to-one. To determine if a function is one-to-one, you must check if every entry in the domain maps to a unique output in the range. This analysis requires understanding the concept of injectivity, which is a mathematical condition where different inputs always result in different outputs.
In the exercise with the function \(f(x) = x^4\), function analysis helps identify potential pairs of different inputs that lead to the same output, like \(2\) and \(-2\). Such insights guide us towards recognizing whether the function's input-output relationship satisfies the one-to-one function criteria.

Mathematical Functions

Mathematical functions are relationships where each input in a set is associated with exactly one output. When discussing specific functions like \(f(x) = x^4\), it helps to recall basic properties:

• The domain is the set of all possible inputs, usually all real numbers for functions like polynomials.
• The range consists of all outputs the function can produce, which, in this case, is non-negative real numbers as the square of a number is always non-negative.

Understanding how each input correlates to an output is key to analyzing whether the function is one-to-one. Since \(f(x) = x^4\) maps both positive and negative inputs to the same output (e.g., \(2^4 = (-2)^4 = 16\)), the function does not meet the criteria for being one-to-one.

Interpreting Function Behavior

Interpreting the behavior of a function involves examining how it behaves across its domain. For instance, we can graph \(f(x) = x^4\) to visually explore how inputs map to outputs. The function's plot reveals that it forms a 'U' shape, where both positive and negative inputs yield the same positive output.
This mirrored symmetry around the y-axis underpins the function not being one-to-one.

• Visual patterns from graphs highlight output repetition for different inputs.
• Such behaviors are critical in distinguishing between one-to-one and non-one-to-one functions.

Insights from graphs and numerical evaluations enhance our understanding of the specific roles particular inputs play in determining output, thereby aiding in concluding the function's injective nature.