Absolute Valued Functions and Their Properties

Absolute Valued Functions and Their Properties

An absolute value function is a type of function that involves the absolute value of the input variable. The absolute value of a number is its distance from zero on the number line, regardless of direction. The absolute value is always non-negative.

The absolute value function is defined as:

$$f(x) = |x|$$

This function outputs the non-negative value of $x$. It can be defined piecewise as:

$$f(x) = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

The absolute value of a number can be thought of as a function that "removes the negative sign" from any negative input while leaving positive numbers unchanged.

General Form of an Absolute Value Function

The absolute value function can also be generalized to more complex expressions, like:

$$f(x) = |ax + b|$$

Where $a$ and $b$ are constants. The behavior of this function can be understood by breaking it down into two cases based on the value inside the absolute value.

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1. Graph of an Absolute Value Function

The graph of an absolute value function is a V-shaped curve. The graph reflects the behavior of the absolute value, which "flips" negative values to positive values.

• For the function $f(x) = |x|$, the graph consists of two rays meeting at the origin, one with a slope of 1 (for $x \geq 0$) and the other with a slope of -1 (for $x < 0$).

Example: Graph of $f(x) = |x|$

• For $x \geq 0$, $f(x) = x$.
• For $x < 0$, $f(x) = -x$.

The graph forms a "V" centered at the origin (0, 0), with the arms of the "V" sloping at angles of 45°.

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2. Properties of Absolute Value Functions

a. Non-Negativity

The key property of the absolute value function is that it always yields non-negative values:

$$|x| \geq 0 \quad \text{for all} \quad x \in \mathbb{R}$$

This means that the absolute value function never produces negative outputs, regardless of the input.

b. Symmetry

The graph of the absolute value function is symmetric with respect to the y-axis. This means that:

$$f(-x) = f(x) \quad \text{for all} \quad x \in \mathbb{R}$$

This symmetry implies that the graph of $f(x) = |x|$ is identical on both sides of the y-axis.

c. Vertex

For the basic absolute value function $f(x) = |x|$, the vertex is at the origin $(0, 0)$. The vertex represents the point where the graph changes direction from sloping down to sloping up. For other functions involving absolute values, the vertex will occur at the point where the expression inside the absolute value equals zero.

d. Piecewise Definition

The absolute value function can be expressed as a piecewise function:

$$f(x) = |ax + b| = \begin{cases} ax + b & \text{if } ax + b \geq 0 \\ -(ax + b) & \text{if } ax + b < 0 \end{cases}$$

For example:

• $f(x) = |2x - 3|$ is defined as:

$$f(x) = \begin{cases} 2x - 3 & \text{if } x \geq \frac{3}{2} \\ -(2x - 3) & \text{if } x < \frac{3}{2} \end{cases}$$

The expression inside the absolute value determines where the function changes behavior. The critical point occurs when $ax + b = 0$, or $x = -\frac{b}{a}$.

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3. Transformations of Absolute Value Functions

The graph of an absolute value function can undergo several transformations, including translations, reflections, and scalings. These transformations allow us to understand how the basic graph $f(x) = |x|$ can be altered.

a. Vertical and Horizontal Translations

The graph can be shifted vertically or horizontally by adding constants inside or outside the absolute value expression.

• Vertical Shift: If the function is $f(x) = |x| + c$, the graph shifts vertically by $c$ units.

  • If $c > 0$, the graph shifts upward.
  • If $c < 0$, the graph shifts downward.

• Horizontal Shift: If the function is $f(x) = |x - h|$, the graph shifts horizontally by $h$ units.

  • If $h > 0$, the graph shifts to the right.
  • If $h < 0$, the graph shifts to the left.

Example:

• $f(x) = |x - 3|$ shifts the graph of $|x|$ 3 units to the right.
• $f(x) = |x| + 2$ shifts the graph of $|x|$ 2 units upward.

b. Reflections

The graph of the absolute value function can be reflected across the x-axis or y-axis.

• Reflection about the x-axis: If the function is $f(x) = -|x|$, the graph reflects across the x-axis, so the V-shape opens downward instead of upward.
• Reflection about the y-axis: The graph of the absolute value function $|x|$ is already symmetric about the y-axis.

c. Vertical and Horizontal Stretches/Compressions

The graph can be stretched or compressed by modifying the coefficient of $x$ inside the absolute value.

• Vertical Stretch/Compression: If the function is $f(x) = a|x|$, the graph is stretched vertically if $|a| > 1$ or compressed if $0 < |a| < 1$.
• Horizontal Stretch/Compression: If the function is $f(x) = |bx|$, the graph is stretched horizontally if $0 < |b| < 1$ or compressed if $|b| > 1$.

Example:

• $f(x) = 2|x|$ stretches the graph vertically by a factor of 2.
• $f(x) = \frac{1}{2}|x|$ compresses the graph vertically by a factor of $\frac{1}{2}$.
• $f(x) = |2x|$ compresses the graph horizontally by a factor of 2.

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4. Solving Absolute Value Equations and Inequalities

a. Absolute Value Equations

To solve an equation involving absolute values, we use the definition of absolute value, which splits the equation into two cases.

For example, to solve $|x - 5| = 3$:

1. Split the equation into two cases: $$x - 5 = 3 \quad \text{or} \quad x - 5 = -3$$
2. Solve each case:
   • $x - 5 = 3$ gives $x = 8$
   • $x - 5 = -3$ gives $x = 2$

So, the solution is $x = 8$ or $x = 2$.

b. Absolute Value Inequalities

When solving inequalities involving absolute values, you again split the inequality into two cases and solve each separately.

For example, solve $|x - 3| \leq 4$:

1. Split the inequality into two cases: $$-4 \leq x - 3 \leq 4$$
2. Solve for $x$ by adding 3 to each part of the inequality: $$-1 \leq x \leq 7$$

So, the solution is $-1 \leq x \leq 7$.

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5. Applications of Absolute Value Functions

Absolute value functions have a wide range of applications in mathematics and real-world scenarios:

• Distance: Absolute value represents distance on the number line. For example, the distance between two points $a$ and $b$ is $|a - b|$.
• Optimization: Absolute value functions can be used to model situations where deviation from a central point needs to be minimized, such as in error analysis or minimizing costs.
• Piecewise models: Many real-world systems, such as motion with obstacles or economics, can be modeled using piecewise absolute value functions.

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Conclusion

Absolute value functions are essential tools in algebra and calculus due to their simplicity and wide application. Their properties, such as symmetry, non-negativity, and piecewise behavior, help us analyze and solve various mathematical problems. Transformations of absolute value functions allow for flexibility in modeling real-world situations. Understanding how to solve absolute value equations and inequalities is a key skill in higher mathematics.