Problem 11 - Algebra for College Students

Question

Determine whether the function \(f\) is one-to-one. $$ f(x)=|x|+1 $$

Step-by-Step Solution

Answer

The function is not one-to-one.

Step 1: Understanding One-to-One Functions

A function is one-to-one if different inputs produce different outputs. Mathematically, a function \( f \) is one-to-one if for every \( a eq b \), \( f(a) eq f(b) \).

Step 2: Analyzing the Given Function

We need to analyze the function \( f(x) = |x| + 1 \). The absolute value function \( |x| \) produces the same output for \( x \) and \( -x \). Thus, \( f(x) = x + 1 \) and \( f(-x) = x + 1 \) have the same output for a positive \( x \).

Step 3: Testing for Distinct Inputs

Select example inputs: let \( a = 3 \) and \( b = -3 \). Compute: \( f(a) = |3| + 1 = 4 \) and \( f(b) = |-3| + 1 = 4 \). Since \( f(a) = f(b) \), this demonstrates that different inputs produce the same output.

Step 4: Concluding One-to-One Test

Since there exist pairs of inputs, such as \( a = 3 \) and \( b = -3 \), where \( a eq b \) but \( f(a) = f(b) \), the function \( f(x) = |x| + 1 \) is not one-to-one.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics. It is denoted by the vertical bars surrounding a variable or expression, like \(|x|\). This function measures the distance of a number from zero on a number line, regardless of direction. For any real number \(x\):

• If \(x\) is positive or zero, then \(|x| = x\).
• If \(x\) is negative, then \(|x| = -x\).

The output is always non-negative, meaning you never get negative results from an absolute value.
For example, \(|3| = 3\) and \(|-3| = 3\). Hence, the absolute value treats numbers and their negatives as equal, at least in terms of their distance from zero. This property is crucial when determining if a function involving absolute value is one-to-one, as in this exercise.

Function Analysis

Function analysis involves understanding how a function behaves, including its graph, symmetry, and whether it is one-to-one. In this exercise, we analyze the behavior of the function \(f(x) = |x| + 1\).
The absolute value component \(|x|\) causes the function to be symmetric about the y-axis. This symmetry implies that for any positive input \(x\), there is a corresponding negative input \(-x\) that results in the same output. The function \(f(x) = |x| + 1\) shifts the absolute value function vertically by 1 unit, but this does not impact the symmetry. Due to this characteristic symmetry, the function does not have a unique output for each input: \(f(x) = f(-x)\). Therefore, this lack of uniqueness in outputs for distinct inputs shows that \(f(x) = |x| + 1\) does not pass the horizontal line test for being one-to-one.

Input-Output Relationship

The input-output relationship in a function describes how each input is paired with an output. In the context of one-to-one functions, each distinct input must produce a distinct output.

• A function is one-to-one (injective) if and only if no two different inputs lead to the same output.
• Conversely, if two different inputs produce the same output, the function is not one-to-one.

For instance, let’s take the function \(f(x) = |x| + 1\) and consider the inputs \(3\) and \(-3\). Both inputs give the same output: \(f(3) = 4\) and \(f(-3) = 4\).
Thus, the points \(3\) and \(-3\) clearly illustrate that different inputs result in the same output. This relationship reveals that the function \(f(x) = |x| + 1\) is not one-to-one due to its ability to map multiple inputs to a single output.