Modulus Functions and Their Properties

Modulus Functions

• Definition: The modulus function, denoted as |x|, converts any input into a positive value.

  • Defined as:

  • |x| = x if x ≥ 0

  • |x| = -x if x < 0

  • Sometimes referred to as the absolute value function.

Sketching Graphs of Modulus Functions

• For the function of the form: $y = a|x + p| + q$

  • The graph resembles a "V" when ( a > 0 ) or an upside-down "V" when ( a < 0 ).

  • Vertex located at the point (-p, q).

  • Roots depend on the positioning of the vertex and the orientation of the graph, which can yield:

  • 0, 1, or 2 roots.

Steps for Sketching Modulus Graphs

1. Graph y = f(x): Start by sketching the original function graph, y = f(x).

2. Reflect: For $y = |f(x)|$, reflect any part of the graph that is below the x-axis about the x-axis.

Key Comparison

• Difference between y = |f(x)| and y = f(|x|):

  • $y = |f(x)|$ does not extend below the x-axis and can appear asymmetric.

  • $y = f(|x|)$ is always symmetrical about the y-axis and can cross the x-axis.

Solving Equations Involving Modulus Functions

1. Sketch Graphs: Begin by sketching the graphs involved, including reflected parts for modulus functions.

2. Identify Intersections: Locate where the graphs intersect to identify solutions.

3. Solve Equations:

   • For equations of the form |f(x)| = |g(x)|, create two cases:

     • Case 1: f(x) = g(x)

     • Case 2: f(x) = -g(x)

Exam Tips

• Emphasize graph sketching to visualize solutions; algebraic solutions may sometimes overlook valid intersections.