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An absolute value function is characterized by the presence of the absolute value operator, which affects how we graph the function. The absolute value of any number is the distance on the number line from zero, always expressed as a non-negative number. For instance, the absolute value of both 5 and -5 is 5.
Functions like \( f(x) = |x - 1| + |x + 1| \) use this property. We identify critical points, where the behavior of the absolute value changes the function - in this case at \(x = 1\) and \(x = -1\). List out:**

• For \(x < -1\), break the function into \( |x-1| = -x + 1 \) and \( |x+1| = -x - 1 \), resulting in \(-2x\).
• For \(-1 \leq x < 1\), the absolute changes direction, giving \(2\) as the function's output.
• For \(x \geq 1\), each absolute value resolves to a positive term, resulting in \(2x\).

Graph these points to reveal a V-shaped graph with a flat section between \(x = -1\) and \(x = 1\). This showcases the step-like transformation characteristic of absolute value functions.

Piecewise functions are defined by different expressions for different parts of their domain. It is important to focus on "breakpoints," where one functional expression shifts to another. This is what differentiates piecewise from regular functions.
An example is \(f(x)\) being defined in two distinct parts:

• For \(-1 \leq x \leq 1\), it is governed by the exponential \(f(x) = 3^x\).
• For \(1 < x < 4\), the expression transitions to linear with \(f(x)=4-x\).

The transition point at \(x=1\) ensures continuity as both segments meet and equal 3. This showcases how piecewise functions allow for distinct behaviors, like exponential growth and linear decline, in different sections of the graph.

The greatest integer function, also known as the floor function, returns the greatest integer less than or equal to a number. It appears as \([x]\) and alters graphs by creating steps, jumping at integers.
Consider the function \(f(x) = [x] + |x - 1|\). This is evaluated in different intervals based on \([x]\):

• For \(-1 \leq x < 0\), evaluate as \(-x\).
• From \(0 \leq x < 1\), the result adjusts to \(1-x\).
• For \(1 \leq x < 2\), the calculation is static at \(1\).
• Lastly, for \(2 \leq x < 3\), compute as \(x + 1\).

These adjustments create a stepped graph, critical for understanding discontinuities and integer jumps implicit in \([x]\).

Polynomial functions feature variables raised to various powers and are described by sums of these terms. They are pivotal in modeling and graphing complex relationships.
Consider a simple piecewise polynomial function like \(f(x)\) with quadratic and linear components:

• For \(|x| < 1\), represented by \(f(x) = x^4\), creating a U-shaped parabola.
• For \(|x| \geq 1\), \(f(x) = x\) presents linear components extending to both sides.

The transition at \(x = -1\) and \(x = 1\) connects these sections, demonstrating polynomial versatility with continuous yet diverse outcomes over its domain. The powerful combination of quadratic and linear components highlights polynomial adaptability in graphing and functional representation.