Absolute Value Function

Definition

The absolute value function is a mathematical function that measures the distance of a number from zero on the number line, regardless of direction. It is denoted as $$f(x) = |x|$$, where the output is always non-negative, making it a piecewise function with two cases: when $$x$$ is positive or zero, and when $$x$$ is negative. This function plays an important role in understanding inequalities, graphing, and transformations.

5 Must Know Facts For Your Next Test

1. The absolute value function is always non-negative, meaning that for any real number $$x$$, $$|x| \geq 0$$.
2. The graph of the absolute value function has a V-shape and consists of two linear parts: one for positive inputs and another for negative inputs.
3. The function can be defined in terms of cases: if $$x \geq 0$$ then $$|x| = x$$, and if $$x < 0$$ then $$|x| = -x$$.
4. The absolute value function is used to express distances, making it crucial in geometry and real-world applications like measuring deviation from a target.
5. In solving inequalities involving absolute values, you often split the inequality into two separate cases based on the definition of absolute value.

Review Questions

• How does the definition of the absolute value function influence its graph?
  • The definition of the absolute value function directly influences its graph by creating a V-shape that reflects the nature of distances. Since it outputs only non-negative values, regardless of whether the input is positive or negative, the graph consists of two linear segments meeting at the vertex (0, 0). This unique shape illustrates how distances are treated equally, which is fundamental when visualizing and solving problems related to absolute values.
• Discuss how understanding the absolute value function can help in solving inequalities.
  • Understanding the absolute value function aids in solving inequalities by allowing us to rewrite expressions in a manageable form. When we encounter an inequality like $$|x| < a$$ or $$|x| > b$$, we can break it down into two cases. For instance, $$|x| < a$$ translates to $$-a < x < a$$, while $$|x| > b$$ means either $$x < -b$$ or $$x > b$$. This approach simplifies complex problems by framing them in terms of linear inequalities.
• Evaluate how transformations affect the graph of the absolute value function.
  • Transformations significantly affect the graph of the absolute value function in various ways. For example, if we consider the function $$f(x) = |x - h| + k$$, where $$h$$ represents horizontal shifts and $$k$$ represents vertical shifts. The vertex of the graph moves to point (h, k), which alters its position on the coordinate plane while preserving its V-shape. Understanding these transformations helps in predicting how changes to parameters will affect graphical representations and can be applied in modeling real-world scenarios.

Related terms

Piecewise Function: A function that is defined by different expressions based on the input value, often used to define the absolute value function.

Graph of Absolute Value: The graph of the absolute value function forms a V-shape, opening upwards with the vertex at the origin (0, 0).

Inequalities: Mathematical statements that compare values, where absolute values often help in solving expressions involving distances and constraints.