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The absolute value function is a fundamental concept in algebra. It is represented as \(|x|\), where \(x\) is any real number. The absolute value function takes any real number and returns its non-negative value. This means that \(|x|\) is how far \(x\) is from zero on the number line, ignoring the direction. For example,

• \(|5| = 5\)
• \(|-5| = 5\)
• \(|0| = 0\)

The function \(y = |x-3|\) shifts the basic absolute value function \(y = |x|\) three units to the right. Any input \(x\) that makes \(x-3\) equal to zero results in \(y = 0\). In this case, when \(x = 3\), the output \(y\) is at its minimum value of zero. Generally, the graph of an absolute value function is a "V" shape, with the point at which the graph changes direction called the vertex.

Real numbers include all the numbers on the number line. This encompasses both rational numbers, like fractions and integers, and irrational numbers, which cannot be expressed as straightforward fractions (like \(\pi\) and \(\sqrt{2}\)). Real numbers are crucial in mathematics as they allow us to represent quantities that occur in real life, from lengths and areas to probabilities. When discussing functions like \(y = |x - 3|\), stating the domain as all real numbers means we're considering every possible real number as an input value for \(x\). Since an absolute value function doesn't create any undefined situations (like division by zero or taking the square root of a negative number), it can "accept" any real number as a valid input.

Functions essentially express a relationship between inputs (domain) and outputs (range). In any function, each input is mapped to exactly one output. For the function \(y = |x - 3|\), the input \(x\) can be any real number, which gets processed by the function to produce the output \(y\).

• If \(x = 0\): \(y = |0 - 3| = 3\)
• If \(x = 3\): \(y = |3 - 3| = 0\)
• If \(x = 5\): \(y = |5 - 3| = 2\)

These examples show how outputs relate directly to their inputs through the function's rule. The range of potential outputs helps us understand what this function can produce, in this case, any non-negative real number. Understanding function input-output relationships is key to analyzing and predicting a function's behavior across different input scenarios.