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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 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.

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.