Absolute value function

Definition

The absolute value function is a mathematical function that outputs the non-negative value of a real number, representing its distance from zero on the number line. This function is denoted as $$f(x) = |x|$$, where the output is always positive or zero, regardless of whether the input is positive or negative. The shape of its graph is a V, which reflects the way it treats numbers symmetrically about the origin.

5 Must Know Facts For Your Next Test

1. The graph of the absolute value function forms a V shape, with its vertex at the origin (0,0) and opening upwards.
2. The absolute value function is defined for all real numbers, meaning it has no restrictions on its domain.
3. When solving absolute value equations, it's crucial to consider both the positive and negative scenarios that arise from the definition.
4. Inequalities involving absolute values can result in two separate inequalities that need to be solved individually.
5. Absolute value functions are continuous and piecewise linear, meaning they can be expressed with different linear equations based on the input value.

Review Questions

• How does the graph of the absolute value function reflect its mathematical properties?
  • The graph of the absolute value function reflects its properties by forming a V shape. This shape demonstrates that for any input value, whether positive or negative, the output is always non-negative. The point where the two lines meet at the origin shows that both positive and negative inputs yield the same output, highlighting how distance from zero is treated uniformly. The upward opening reinforces that there are no negative outputs in this function.
• What steps should you take when solving an equation involving absolute values?
  • When solving an equation with absolute values, first isolate the absolute value expression. Next, set up two separate equations: one for the positive case and one for the negative case. Solve both equations independently to find all possible solutions. Finally, check each solution in the original equation to ensure they satisfy it since extraneous solutions may arise from squaring or manipulating inequalities.
• Evaluate how absolute value functions are used to model real-world situations involving distance.
  • Absolute value functions are particularly useful in modeling real-world situations involving distance because they measure how far a point is from a reference point without regard to direction. For example, if you're analyzing how far a car is from a certain location, you can use an absolute value function to represent that distance. This property allows for more straightforward calculations in cases like determining the proximity of different objects or understanding fluctuations around a central point. Such applications underscore how essential absolute values are in various fields like physics, engineering, and economics.

Related terms

Inequality: A mathematical expression that shows the relationship between two values that are not equal, often involving symbols such as $$<$$, $$>$$, $$\leq$$, and $$\geq$$.

Piecewise Function: A function defined by multiple sub-functions, where each sub-function applies to a specific interval of the input variable.

Distance: The absolute value can also be interpreted as the distance between two points on a number line, without regard to direction.